Just tighten it down more. You won't hurt your frame. All you are doing is compressing your rear wheel axle, which is harmless. Make sure the QR ends are centered in the mounting cups and the QR lever isn't interfering with the mounting cup.
Also, just say'n, but sometimes cheap things don't work as intended. It is quite possible the mounting points are loosening due to vibration as you are using the trainer. I had a Blackburn Trakstand (Blackburn's cheapest fluid trainer) where the screw tightening the resistance unit against the tire would loosen as I rode. If this is the case, perhaps a little blue locktit might work.
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Cat 2 Track, Cat 3 Road.
"If you’re new enough [to racing] that you would ask such question, then i would hazard a guess that if you just made up a workout that sounded hard to do, and did it, you’d probably get faster." --
the tiniest sprinter