I believe there is a glitch in the Tsukuba track challenge here.. at the first split time you're only about 15 seconds behind, but at the next, you're suddenly about 30 seconds behind.. In all the other challenges you start out being about 15 seconds behind, but there is no problem winning those challenges being about 10 seconds ahead..
I tried tuning the **** out of a '89 Miata and never got a faster lap than 1.07.996 on the Tsukuba track, but had no problem beating the other challenges, therefore my thoughts are, that it's just a bug in the game..
But i guess i have to beat the Tsukuba challenge using that Mugen to get the win.. even though i consider it cheating to use racecars to beat regular cars, but hey, Life's not fair