You could get the frame painted in the predominant or background colour and then do the design over that.
Unless you already have experience of doing designs on 3D surfaces, a bicycle might be too tricky to start on. Only weeks ago you could buy a perfectly good small retoucher's airbrush kit, complete with compressor, at Lidl for 60 euro; they even gave one paints in the kit. I saw just an airbrush (better than the one I bought at Lidl), no compressor, at Banggood the other day for 13 Euro delivered.
But the problem isn't the equipment, it is the skill. If I were doing this job, I'd have the bike professionally sandblasted and painted, and then make and print out the design life-size on frisket (self-sticking masking and cutting paper used in airbrushing) in sections with jointing crosshairs (QuarkXPress and other layout programs add these automatically if you ask them to). Then I'd hand bike and 1:1 art on frisket to an airbrush artist with hotrod experience, rather than the more common photo retouchers (the latter are used to working on flat images). This person will very likely ask you for another or several more sets of the printouts, if your design is at all complicated and needs to be applied in layers. If you can't find an airbrush artist, ask at the local shops who painted their signs, and get the signwriter to do the job (generally with brushes) in a waterproof paint he, she or it has experience of, very likely acrylic.
Then take the bike back to the first car duco expert and get the whole thing clear-coated with some non-yellowing lacquer. Acrylic paint is plenty tough, but you don't know how thick the airbrushed put it on, except that the more expert your painter is, the thinner the layer of paint will be. So, lacquer or some other poly-something protection.
Of course, if you have experience of painting designs on small tubes, ignore this and go for it.
In either case, don't forget to send photographs for us to perve over. Good luck.
