Cork is rather bad for underfloor heating. If necessary, the underfloor heating can also be installed in the walls, but I would probably only go to that effort with really high-quality wooden floors.
There are huge differences in the feel and durability of vinyl that cannot be determined by price. There are very large discounts online, and you can order samples.
I also have vinyl, am satisfied, but would only install vinyl with the Blue Angel label. I find glued vinyl significantly better, especially on the ground floor. Gluable vinyl with the Blue Angel label has only been available for about a year (from Meister and, if I remember correctly, also from Wineo).
Almost all design coverings are only click systems and have corresponding carrier layers. The coverings from Classen have the best suitability for underfloor heating among click coverings.
It is worth taking a look at the datasheets regarding thermal conductivity and abrasion class.