In that case I'll share some info on my attempts,
I've been trying the Daihatsu OFC-1 and can't get under 8 minutes (let alone 7"35"000)- the thing just doesn't have enough power (only 160bhp or thereabouts fully modded except using Mid-rpm turbo). It carries the speed well but just doesn't have enough outright speed- I can only get up to about 130mph on the long straight (even with higher gearing).
If you do try this car, though I'd advise against it, watch the high apexes as on my laps with it I nearly rolled it multiple times
Admittedly I'm not the best driver and am only on a DS3 so I reckon a decent driver with a wheel could possibly creep under 8 minutes.
I'm looking out for some Kei cars to generate in my UCD but as of yet- no luck