That's not likely — quoted camera draw is 90ma and the reverse light circuit is quite capable of running two lights, which would pull about 4A. Quite a few VW owners have converted the rear fog to a reverse light by simply connecting the two lamps together (and removing the red lens, of course.)
I'd expect it to still work even with two reverse lights connected — the relevant fuse is probably 7.5 or 10A.
Does the reverse light still work when the camera doesn't? My thought is that you may have it wired to the wrong terminal — check it with a multimeter.
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