So, I just tried this out, and it feels like it really was a setup issue.
I first ran time trial in Melbourne, trying to find viable setups. And after some time, I actually found one, for both the dry and the wet conditions. I even improved my dry PB by a second with the new setup.
The key to using TT setups in Career Mode races with enabled carcass temps is to minimize the tire pressures, as otherwise, your tires will overheat like crazy. It might decrease tire life, but it's the far safer option.
And believe me, while this worked as expected, the ridge between racing and spinning out is very limited. The chance of spinning out if you take it too aggressive is high, so easing throttle and steering is the best you can do, especially when your tires are approaching the 60%. Overall though, I had a much easier time with the new setup, as I managed to get through all the practice programs after getting used to the setup.
If you want to know what setup I found for Melbourne, maybe I'll make a follow up reply in which I show you what I came up with.