When conti says 35 it means 35 total outside dimension. They include all rubber. So if you can fit 32 with some room, maybe the 35 will fit. The Contact Travel 37mm are the same as many of the 33mm cyclocross tires out there.
I've had 32mm Hyper on there and they're ok but there's not a great deal more room. It will depend on how the tyre casing inflates. I've got some 35mm Hypers I could test fit. Actually, perhaps they would be ok for TCR?
Never heard of them
https://thesuperiorbikeshop.com/continental-gp-urban-tire/
35mm is probably too big for my frame though.