To clarify:
Gear and Avatar's Favor can lower perpetuation down to a minimum of 1 mp/tick. Once it hits 1 mp/tick, further -perpetuation gear does nothing. It remains 1 mp/tick.
Auto Refresh's 1 mp/tick gain will be consumed by that, which will give an avatar with 0 perpetuation cost. When you get Auto-Refresh II (2 mp/tick) you will now gain 1 mp/tick while an avatar is out.
Refresh gear (and effects like the spell and ballad) stacks on top of it. If you haven't capped -perpetuation, refresh simply acts as -perpetuation. If you have, it is simply added to total refresh.
Titan, Ramuh, Garuda, Leviathan, Shiva, Ifrit, and Diabolos all cost 15/tick perpetuation at 99. Fenrir costs 13, and Carbuncle is 11.
Avatar's favor lowers the damage (physical and magical) your avatar does quite noticeably and resets when you bloodpact anyways, so I avoid using it (wish they would change this...)
So you want to shoot for -14 perpetuation cost if you can, and add some refresh on top of that.
Ultimately (if you don't have something shiny and rare like Nirvana) you will want to make a Magian Staff for Avatars you use a lot (e.g. Shiva, Garuda, Ramuh). This is -7/tick
Evoker's Ring is a staple, it has -1/tick (8 total)
Caller's Doublet +2 has -4/tick (12 total)
Caller's Pigaches +2 has -3/tick (15 total, 1 past cap)
Auto-Refresh 2 eats the 1/tick perpetuation, and you have 1/tick with your avatar out.
You can add in something like Nares trews for an additional 1/tick (2/tick refresh)
Caller's Horn+2 for additional 2/tick (4/tick refresh)
Swap Caller's Pigaches +2 around with Adhara gages (-2 perpetuation) and serpentes set (hands/feet) for an additional 1/tick (5/tick refresh)
Moonshade earring if you are strictly mages and don't have a tp bonus earring 1/tick (6/tick refresh)
Still a few other options as well. If you don't have Adhara gages but have the other stuff, sitting on -15 perpetuation you can swap Caller's Doublet +2 for Summoner's Double +2 during perpetuation (-14, still capped), as the +crit will benefit your avatar auto-attack more.