If a simulacrum is polymorphed through True Polymorph into a creature that can regain hit points normally, can the simulacrum regain hit points while polymorphed in that way?
Yes, and no.
The simulacrum's statistics are replaced. This means you take whatever form of record sheet you're using for the simulacrum, put it aside, and grab the stat block of the creature you've polymorphed it into and use that while the polymorph effect persists.
The target's game statistics, including mental ability scores, are replaced by the statistics of the chosen beast. It retains its alignment and personality.
While polymorphed, the creature is a tyrannosaur, a giant ape, or what have you. For the duration of the polymorph effect, the simulacrum effectively does not exist, so none of it's restrictions apply to the new replacement.
A true polymorphed creature can regain hit points through resting (polymorph doesn't last long enough to complete a short rest to use hit dice, let alone a long rest). Using true polymorph also lets you potentially pick something with regeneration, a feature not found on any beast I'm aware of.
The target assumes the hit points of its new form. When it reverts to its normal form, the creature returns to the number of hit points it had before it transformed.
When the polymorph effect expires and the creature resumes being a simulacrum, it has however many hit points it had before the polymorph. Of course, that's how it works for any other creature, too.