The Life Cleric has 2 separate features which you've conflated here. The first one is Disciple of Life.

Whenever you use a spell of 1st level or higher to restore hit points to a creature, the creature regains additional hit points equal to 2 + the spell’s level.

The second one is Blessed Healer.

When you cast a spell of 1st level or higher that restores hit points to a creature other than you, you regain hit points equal to 2 + the spell’s level.

Blessed Healer is simple. You get healed when you cast a spell. So Regenerate will heal you for 9 hp, but only when you cast it.

Disciple of Life is a bit less clear. It is triggered "when you use a spell to restore hit points to a creature". There are 2 interpretations of this:

1. The first is that using the spell is the same thing as casting it, and therefore only the initial 4d8 + 15 gets the +9 hp from Disciple of Life.
2. The second is that you're using the spell to restore hit points every time it restores hit points. In that case, the target would indeed gain 10 hp every turn for the duration.

Unfortunately, there is basically nothing in the rules to help decide which interpretation is correct. "Using" a spell is not common terminology - almost everything in the rules talks about casting a spell, not using it.

However, a similar question was answered in the official Sage Advice Compendium.

If I’m a cleric/druid with the Disciple of Life feature, does the goodberry spell benefit from the feature? Yes. The Disciple of Life feature would make each berry restore 4 hit points, instead of 1, assuming you cast goodberry with a 1st-level spell slot.

If Disciple of Life works every time someone eats a goodberry, it should certainly work every time someone is healed by Regenerate. So yes, Regenerate cast by a Life Cleric will heal 10 hp every turn.

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.

RAI agrees

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.