Yes, the simulacrum can regain hit points using hit dice.
The simulacrum spell description says, in part:
[...] The duplicate is a creature, partially real and formed from ice or snow, and it can take actions and otherwise be affected as a normal creature. It appears to be the same as the original, but it has half the creature's hit point maximum and is formed without any equipment. Otherwise, the illusion uses all the statistics of the creature it duplicates, except that it is a construct.
The simulacrum is a creature, and creatures regain hit points via hit dice. As noted in the quote above the simulacrum uses all the statistics of the original, except it has a different hit point maximum and has no equipment, neither of which prevent it from using hit dice.
Furthermore the spell says the simulacrum "lacks the ability to learn or become more powerful, so it never increases its level or other abilities, nor can it regain expended spell slots". Using hit dice is not learning, increasing a level or other ability, or regaining an expended spell slot.
Furthermore, the spell says "if the simulacrum is damaged, you can repair it" in a lab. It does not say you MUST repair it in a lab, and it doesn't say you can ONLY repair it in a lab. Without MUST or ONLY, the plain English reading of this sentence is that it is an additional way for the simulacrum to regain hit points, but not the only way.
Furthermore, simulacrums are constructs, and generally constructs cannot be healed, but they regain hit points during rests.
Finally, there is this November 2016 tweet in which rules designer Jeremy Crawford was asked:
Can the creature created by Simulacrum be healed by spells?
To restore hit points to a simulacrum, you must use the costly alchemical procedure mentioned in the spell.
I note that JC was asked about healing a simulacrum with spells, in which he cited that you must use the alchemical procedure. That can be read strictly that the response was in answer to the question, or, it can be read that JC was making a global statement regarding all the ways a simulacrum can regain hit points.
I argue that his statement was not global, referring to all possible ways of restoring hit points to a simulacrum, because if it was, it means that a simulacrum could not regain hit points by means of wish or divine intervention. He could have easily tacked on the end, "barring wish or similar". Statements that are absolutes usually say "by any means" or similar language, such as the limit of abilities to 30. So that means we have to read into the tweet that he WAS excluding hit dice, but was NOT excluding wish, which seems like a lot to read into a tweet.
Obviously, this is interpretative. In the end, the GM is going to have to adjudicate.