Maybe invest in a battery maintainer that you can leave hooked up to the battery if car is in garage or close to house off road, failing that try a solar charger that can be left on the dash which would trickle charge the battery.
I'll second this, I too was mightily impressed with the UX whilst I had it as a courtesy car think I would be I clined to wait though for new model year with the updated dash etc.
I'm 58 with 25yrs+ ncb kent post code last year my GS-F with AA insurance I paid £424 this year rocketed up to £760 needless to say I've shopped around and managed to find insurance same as last year' cost albeit had to suffer slightly more compulsory excess but still with zero voluntary excess and with up to 7000 mls per annum (it isn't my daily so probably could have less miles) it does pay to shop around.
I applaud you for your patience in persevering with trying to get this engine/fuelling problem sorted, but I have to ask after all this would you, once car is sorted, be really comfortable/have confidence with the vehicle going forward ?.
Sounds like the 12v battery is flat or too low a voltage to boot the computers, try connecting a 12v power source (lithium booster pack or a 12v battery via jump leads, doner car need not be running as its only the 12volts needed to be seen by flat battery) to the auxiliary 12v battery in the boot the car should then start up.
My relax warranty automatically kicked in (sent email) when my manufacturer warranty expired on the overlap from when I had my gs f serviced last year and as far as I can see so long as the to the service timescale/interval is observed the warranty wouldn't lapse surely.
Sidcup Lexus is also Jemca and they've always been good to myself for the last 4 years I've taken the GS there, saying that though Lexus Maidstone are good but bit of a distance from the Oval
Surely if the car is under a service plan it would be via Lexus approved agency not solely for your dealership and could be used at any other dealership or does each dealership have their own credit service facility?