I have a sort of cheap/ worn out trainer. I actually use a mavic QR in it and I clamp it down to the point where I need to put my body weight onto the locking lever to get it to snap in. I have never damaged my carbon framed bike (Knock on Carbon).

In any case I would say just crank it down more, problem with your trainer is that both sides seem to screw inwards to lock the thing, this makes it harder to get sufficient clamping force. Sort of like if you had a skewer where both sides could only be tightened with your fingers, no QR lever, no alan bolt to put some torque on with, just finger tight. That is basically how your trainer seems to be designed.

I would say look for another trainer on craigslist. Lots of people sell them for around 100$ and alot of times they are actually pretty good trainers, plus you can atleast try locking your bike on it.