Yes, you can
This feat only affects spells with the cold descriptor.
The text doesn't say that you can't prepare a Rime Fireball, only that it won't be affected.
It is perfectly legitimate for your player's Wizard to prepare his Rime Fireball (with the required +1 level increase). During casting, he'll have the option to either:
- Expand one of his Admixture daily uses to change it to a Cold spell, benefiting from the Rime Spell entangling effect.
- Forgo the Rime Spell effect. and cast it as a normal fireball (or even as an Acid or Shocking spell, which may be useful against some exotic enemies or in special circumstances).
If he chooses the second option, the result is the same as if he simply used a 4th level spell slot to prepare a 3rd level spell - which is perfectly legitimate by the rules anyway.
Further consideration: The interpretation mentioned at the end of your question - stating that
Versatile Evocation modifies the spell after it as been cast while Rime Spell only works while it is being prepared.
(quoted from the question, not the rules)
This may lead to paradoxical results if used the other way around:
If Rime Spell is only concerned with the Cold descriptor during spell preparation, a wizard could prepare a Rime Cone of Cold, and later on, use Admixture to cast it as a Fire spell - resulting in a cone of flames with "frost that clings to the target, impeding it for a short time." - which is thematically ridiculous.
Side Note: Your player's wizard can always use a Metamegic Rod of Rime Spell to apply this combo on-the-fly - this is relevant even with the harsher interpretation stating you can't prepare a non-cold spell with this feat.