It's a shrewd guess, but as you suspected neither of these sentences is idiomatic.
This sort of ellipsis (called Whiz-deletion, a jocular contraction of wh- is) has an ancillary "rule": if only a single adjective is left after the deletion it must be moved in front of the noun:
∗He solved a problem
which was difficult ⇨ okHe solved a difficult problem.
But you can't do this with him, because personal pronouns cannot take attributive adjectives (except in certain exclamations):
∗The cake is eaten by him happy ⇨ ∗The cake is eaten by happy him.
It should be pointed out that neither of the sentences you put forward is natural in English, with or without the deletion. The problem sentence would never be uttered, because the natural way of saying this is with the ordinary preposed attributive: He solved a difficult problem. And him who is happy is a very literary construction; the happy man/guy/fellow is the natural way of saying this. Better examples would be something like these:
The cake was eaten by someone
who was happy to find it.
He solved a problem
which was too difficult for me.
Moreover, it is not clear that the himself quadriplegic phrase at the end of the first sentence should be regarded as a relative clause. It is clearly a supplement or afterthought, and I would be happier understanding it as an appositive.