It is claimed that "Luxury" vinyl is pet friendly. It has low conductivity compared with other hard surfaces, so it might feel a little less cold if you decide to go without sub-floor heat. Still cold compared to carpet though.
Do you clip your dogs' nails? That might solve one of your problems. We never did but the wear of one dog was negligible by comparison with 9 kids.
Have you considered radiant ceilings? They can heat the floor by radiation and are also good with a low-conductivity floor covering. For a new build it seems to be worth consideration. This is theory only: I'd make up metal suspended heating panels rather than try to drive the heat through a conventional ceiling, which should allow a lower water temperature.
Radiant ceilings can also be very effective for cooling, but only if the internal humidity is kept low, or you would get condensation on the ceiling.
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