Another Brit here. I did a west to east Southern Tier in 2013. Started mid March. Temps were not a problem anywhere. A few hot days in the first week. Start early drink plenty fluids. Below freezing overnight in Silver City, New Mexico.
Bikes? Take your own. Honestly. A comfortable bike which you have confidence in is the most essential item. If you aren't confident in reassembly there are plenty bike shops in San Diego.
For getting it there. On British Airwats a boxed bike counts as a suitcase. So free on Premuim Econmy or better. In cattle class costs you about £70 each way. A big cardboard box is all you need. Disassembly required?
What I do is remove saddle/seatpost, pedals, bottle cages, and front rack. These go in my case to ensure boxed bike is under 23Kg.
Thereafter front wheel comes of and is cable tied to front triangle. Bars off and turned sideways. Cable tied in place. Reomove rear derailleur and either pack seperately or cable tie to frame. Cable tie pipe lagging to frame tubes. Re-enforce corners of the box with rolled up cardboard tubes. Use a 100mm length of tubing to put between front fork dropouts with QR to protrect them.
Cable tie everythng so it all comes out the box in one piece. That way if inspected nothing gets lost.
Biggest decision, I would say is how light to go. I use 4 panniers and no bar bag. But I carry a small laptop, tent, sleeping gear, cooking gear etc. If you are prepared to go light two rear panniers and a bar bag is workable. Even for camping. Lightweight tent. No laptop. Ditch cooking gear.
Once you ditch the front panniers you are probably saving 5Kg. 2Kg for the panniers and rack and at least 3 Kg for the stuff you aren't putting in them.