No, the alchemical process is the only way it can regain hit points
If the simulacrum is damaged, you can repair it in an alchemical laboratory, using rare herbs and minerals worth 100 gp per hit point it regains.
Repair not heal
First note that this does not say heal it says repair. Though treated as a normal creature it is still a being made of ice and snow. It is that reason that it likely has a special way to repair it.
Allowing other forms of healing means that the alchemical process would be useless
This statement must preclude other forms of regaining hit points otherwise it is meaningless. Even though it does not say so explicitly, the above method is clearly meant to be the only way to regain hit points for a simulacrum.
It does not say that potions or healing magic or rest doesn't work for example, but if they were possible then the expensive complicated process for healing would be meaningless and there would be no point in including it. Sleep especially makes this meaningless because it is free and available to every creature/class and allows for healing completely. Why would anybody ever pay 100gp per hit point if they could sleep or use any of the myriad of other methods for regaining hit points available? This logic also applies to traits or features that let you heal.
Jeremy Crawford supports this logic as well:
To restore hit points to a simulacrum, you must use the costly alchemical procedure mentioned in the spell.
Duplicated creature must be humanoid or beast
Also note that the creature you are duplicating must be a beast or humanoid and I could not find one of either in the Monster Manual that has regeneration. There may be ones from other sources, but it is something to be aware of.