EOS Alucard shouldn't have regen, due to only having one life. He is omnipresent, though.
But doesn't Level 0 Alucard have notable regeneration feats even with one life? After he emptied all of the souls inside him except for his own and turned them into his familiars:
1. Regenerated from decapitation by Nail Anderson who used a holy bayonet.
2. Regenerated from being vertically split half-way down his body by Nail Anderson using an extremely holy bayonet in hand with a massive amount of holy flames emanating from the blade and the thorns that roasted millions of familiars, after which Alucard turned to another form and killed Anderson.
3. Regenerated from being horizontally severed in half by his giant hellhound and his right arm I believe cut to pieces by Vampire Walter, in which he shifted to the form of a girl.
The souls are used for what I rather call as a resurrection ability, or regeneration for when Alucard is killed. For example, when Alucard is hit by a holy bayonet at the heart (which obviously will kill him), the damage (and this is just a guess) is diverted to one of his souls, and then he regenerates. In normal cases, Alucard is just hit by a powerful attack but that attack is not powerful enough to kill him, but severely damages his body, which he'll regenerate from easily without any loss of souls. So using the souls is only when he's killed (such as when he reformed a new body from bats after multiple stabbings to the heart and decapitation by Anderson) and all other cases is just him regenerating as normal.
I also don't see why his souls are a source of all his regeneration when much weaker vampires who used Alucard's blood to become vampires and always had only one soul/life have regeneration abilities (Luke Valentine being most notable, pre-full vamp Seras given the damage she sustained from Anderson no longer affected her, and any artificial vampire in Millennium). It's a general ability for a vampire to regenerate (albeit low-level) and Alucard should not be an exception given he's the most powerful.
EoS Alucard has two lives actually since he absorbed Schrodinger and added it to his own. It granted him omnipresence/high-level teleportation as well as a high boost to his regenerative-related abilities.