grammar - Is this contraction of 'there is' acceptable to native speakers of English?

It's bad. I've seen the badness attributed to the gap following the contracted form -- "... there's [gap] to know", where the gap is created by the relative pronoun that is removed from this place. Naturally, that came from the MIT direction, since it doesn't make sense. My theory about this (and probably other people's) is that the problem is losing a stressed vowel due to contraction. We start with "... there is something ] to know", where stress comes at the end of the constituent indicated by the right bracket in my schematic representation. If it weren't in a relative construction, that would put stress on "something", and the preceding "is" would not be stressed, so that we could contract the "is" and wind up with " ... there's 1something ] to know". No problem.

However, the "something" is relativized and lost, leaving " ... there 1is ] to know, and now the "is" is stressed, since it has come to be at the end of a constituent. That prevents the vowel being lost, so you can't contract.

I think this is a pleasing theory, since stress on a vowel preventing its deletion seems to me to be a natural constraint on contraction.

Unfortunately, I recall from when I was working on this, there are other examples which seem to favor the gap theory I mentioned above. I can't recall crucial examples right now, but maybe you can think of some.

Also, in some other cases it looks like you can contract a stressed vowel, and the stress on the vowel just gets moved onto a contiguous surviving syllable. "No, I cannot go" -> "No, I can't go".