I think it's a go for this route - probably for next week. Thanks
@DCLane for discovering it for me!
I found a bike shop that hires out dropbar gravel bikes - ideal for this route. If they can drive the bikes (and us) to Stranraer then collect the bikes from Eyemouth, that would be ideal. As far as I can tell, they're near Lockerbie, which is reachable by a direct train from London.
There seems to be quite a lot of rain and wind there at the moment, but you can't have it all (and at this point I'd much rather have that, than 33 Celsius in the shade).
I think I just need to figure the accommodations. Still not sure whether it's worth the effort to lag the camping gear - something to dig into (advice and recommendations of accommodations are welcome).