No to the woodchips bothering the cat. His paws and his weight are not the
same as your feet and your weight. The fact that you can see cats like
walking in it (and using it for a literbox, too, no doubt) is a good
indication that cats are fine with it.

No to putting a heavy, unstable ANYTHING on top of woodchips. Go to ground
level, put down a paving tile, level it, put the birdbath on top and rake
woodchips around it. If it still tips easily, get a plastic one or risk
finding a cat with its brains bashed out someday.