Are there any LFSR-based ciphers with variable feedback polynomials?
I think the reason this is not so popular is a combination of the three reasons you surmised. It is quite difficult to control properties of such a generator. There was at least one paper (can't recall reference) where the suggestion was to use a secret key to make a choice out of a collection of primitive LFSR polynomials.

I do know of another unpublished example, but I need to find my hardcopy of the manuscript to give you the gist of the design, which may take a day or two. It was never submitted for publication, to the best of my knowledge.