If you adhere to the flavor where wavefunction is not real and collapse is just an information update, you can't avoid issues with the early universe: as 'wavefunction is underfined without measurement' (c) Calrid, and the very first measurement devices were possible only when Universe had cooled down enough so the information could be preserved in some system and there was a possibility for the systems, isolated from the environment to some extent, then the whole history of the early Universe appears to be non-realistic. Wavefunction was not defined at all in early Universe (?) until the very first measurement device had formed (magically from non-existent wavefunction?) when (?)
In a flavor where wavefunction is real, you have issues with relativity, described in the original post.