The anime adaptation is pretty good, honestly, but it's nothing compared to the experience of the game. That said, if getting the game is an option (be it the original PS2 version or even the soon-to-be-localized Vita remake), do so, but there's no reason not to check out the anime adaptation.
Since Persona 3 characters are featured as well (namely Aigis, Akihiko, Mitsuru and Elizabeth), I also recommend getting Persona 3 FES from the PlayStation 2 Classics on the PSN ($10, which is honestly a serious bargain since a physical copy of Persona 4, new or used, STILL goes for $40, and Persona 3 Portable is $20 on the digital store right now).
As for Shadow Labrys, most of the damage comes from simultaneous attacks in her strategy (and Asterios can act as a defense, too). It's not difficult to manage both her and Asterios at once knowing their attacks can be synchronized well. If you don't end up using Personas much (as in relying on regular attacks), the experience is really not much different from using the real Labrys.
I can't say much more about Elizabeth, though, since I don't have as much experience with her as my own character (as I'm not a Persona-centric player; she doesn't fit me too well), but I can guarantee Thanatos will dish out serious damage.