A few points:
1. All my UK cycling in the summer half of the year is in sandals with no socks / socks / waterproof socks depending on the weather. Drying socks is easier than drying shoes. I've got some thin waterproof overtrousers. Full length if wet and cold plus a pair that are cut off below the knee for warmer weather and are easier to put on and get off.
2. A repellant called "Smidge" https://www.smidgeup.com/ is easily found in Scotland. Perhaps it is very effective as I've not encountered the little beasts but they don't like rain and wind any more than cyclists. There's a midge forecast on that website.
3. Some (all?) of the Edinburgh - London trains have bike storage which involves hanging the bike by one wheel on a hook in a cupboard. It's not very touring bike-friendly, particularly if you have flat handlebars. I suggest that you read through the discussion at https://forum.cyclinguk.org/viewtopic.php?t=158863. 
4. You've chosen a challenging route north of Perth which is fine if you like big hills. There are alternatives which are longer but easier on the legs. You might be able to see the national cycle routes at https://explore.osmaps.com/find-routes if you turn on the cycle route layer. The alternative I would suggest is: Perth - Pitlochry (route 77) then route 7 to Inverness (two long but not steep hills - that's the route the trains use - and much of the cycle route is on what used to be the road which has been replaced). The route that I've been on north of Inverness is via Alness, Lairg and Bettyhill then along the north coast. South of Bettyhill, turn right at Altnaharra onto the B873 / B871 which is a quiet road. North of Lairg the Crask Inn https://www.thecraskinn.com/offerings is a welcome stop in the middle of nowhere but check in advance if it is open. Last year it was closed on Mondays so a fuel stop in Lairg was needed.

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Right now my quandary is the train luggage allotment...

George, one solution might be to rethink your load and take much less cargo and luggage. The UK has many shops for food resupply and despite a good possibility of rain, you'll be traveling at a warmer time of year....

My trip a year and a half ago (part in USA, part in Ontario Canada), I anticipated warm weather, I wanted to be able to bike in rain without rain pants.  For that trip I brought sandals to wear on the bike to keep my bike shoes dry in rain.  But this trip, I am assuming if I am biking in the rain, I will be wearing rain pants.  And that means rain covers for my bike shoes.  But will probably bring sandals, just in case, it depends on my weight when done packing.  Third photo, the sandals I biked in on rainy days on that trip, that trip was on my titanium bike.  They have a stiff enough sole I can ride all day without foot pain, but I need the right pedals for those sandals to work well.

When I went to Iceland, I only took one jersey, it was long sleeve.  I checked out the weather averages for this trip, I am leaning towards one long sleeve and one short.  Otherwise, pretty much teh same clothing as I took to Iceland.  But I might bring one short and one sleeve shirt instead of two long sleeve.  For biking, am leaning towards two pair of bike pants, one would be shorts and the other would be long pants that convert to shorts.  That is what I used in Iceland.

I always bring a down vest and stocking cap on trips.  Even if it gets quite chilly, a rain jacket (in dry conditions) over the vest is almost as good as a winter jacket.  I looked at the June data for London, Leeds, Edinburgh, and John O Grotes on:
https://weatherspark.com/m/147876/6/Average-Weather-in-June-at-London-Heathrow-Airport-United-Kingdom

Where possible I always look at airport weather data since they have the best data on the past for averages and statistics.

I do not recall where i heard this, but I heard that Scotland can be buggy.  I plan to treat my clothing with Permethrin and also bring insect repellant.  Not sure if I will have a head net or not.

First attached photo is my Nomad Mk II and my Ortliebs, plus handlebar bag.  This would be what I bring on this trip.  The Rack Pack (31 liter duffle on top of rear panniers) would be close to empty with no food.  I do not see myself cutting back much on my load.  And I was thinking that I would buy very little food before I get onto the train to make sure that I do not start out with too much weight and bulk.  At the time I took this photo, I had not yet learned how to correctly attach Rack Pack with Backrollers.

I have both Carradry panniers and also the Ortliebs.  At this time, planning on the Ortliebs, but might change my mind on that later.  Second photo, Carradry in back on my Nomad.  The Carradry are about 25 percent larger than the Ortliebs but weigh about the same.  If I brought the Carradry, would bring a dry bag for the rack top bag.

I have a Carradice Camper, bought it a few years ago.  One day I noticed that the US Dollar was unusually strong, tried to figure out if there was anything that I might want some day, bought the Camper at that time.  I have not used the Camper yet, but I figured if I ever did a credit card tour, I would use it then.  I like the 31 liter Ortlieb, it attaches on the panniers very well, mine weighs 795 grams.  I have not weighed the Camper, the Nelson Long Flap is 1025 grams.  How to correctly attach Rack Pack with Backrollers at this link:
https://www.youtube.com/watch?v=OtxsoOa2h48

If I decide to bring the Carradry instead of Ortliebs, I will consider the Camper.  I used a Nelson Longflap on my Pacific Coast tour, but buckling and unbuckling it from my bike every day was a lot less convenient than the Ortlieb Rack Pack. 

I have a few other ideas for carrying my luggage in the train station, but it is all contingency so I won't elaborate here.  My plans are to start planning with worst case scenario, and later if I find something smaller and lighter will work, do that instead.  But plans start with stuff that I have certainty will work, modifications come later.

At some point I will be trying to remove one or two more pounds (or kg), but that step comes last.

I am assuming food will be readily available south of Leeds, but once north of there I anticipate carrying more food with me, especially north of Edinburgh.  I probably will bring a few days of food from home, in original store packaging, this would be lighter dehydrated stuff that might turn into contingency foods later in the event that I don't plan my grocery shopping well enough.

The part that I think will be hardest is that for my 72 years of existence, all driving is done on the right side of the road.  My helmet mirror is on the left side, it can't be changed.  So, I bought a mirror that attaches to eye glasses so I can try to put that on the right side of my head.

Thanks for sharing this, Dan. I've been thinking about doing a tour of Scotland again but this time with a much lighter load. I've got a Camper Longflap and an Ortlieb bar bag too. On my last Scottish tour I took a lot of gear with me and tbh it was a bit overkill. That said the weather at that time of year was quite tough. Orkney was very wet and heading south to Lairg, from Bettyhill through Altnaharra was horrible!  Now, with the winter behind us a lighter load seems quite possible.

[You're welcome, Ian! I fixed your photo so the inverted thumbnail would display correctly. -- Dan.]

My limited experience of UK railways is that they are a bit more flexible than airlines!  I would book train tickets away from rush hours…….perhaps in the week, in the middle of the day, and not at weekends when there may be sporting events  …..so that the train is not overfull.  I would then expect you could get extra baggage on.

I was very happy with GWR who honoured my 2 week out of date ticket from LHR to Taunton recently after the Iran War delayed me in Sri Lanka.

I don’t know the arrangements London to Edinburgh but a lot of trains now have bike storage areas at the ends of the coaches that you sit in so you load the bike in there yourself and can see it.  Longer distance trains may still use the guards van?

Good advice PH.
Would just add:

- Scottish Ferrys seem to be the news at the moment due to cancellations.  Hopefully resolved by the time you are here, but worth considering options if you plan to use them.
- Nessie photographs best from his right hand side…….