For some reason I never looked at Mihawk's turn counter. I just assumed his barrier was 99 turns. I'll also switch out Shiki, thanks.
I have no clue what to do for powerhouse though. I need to look at the details of that one.
A F2P team that I found on Reddit is 5+ Sanji with friend Legend Blackbeard, Story Magellan, Gasparde, Raid Kuma, and Colo Vista (with those latter two on the bottom row). You get matching orbs on Sanji and BB early on while stalling, use Vista on stage 4 to give you two more matching orbs (and make sure nobody has a non-type orb) then use Sanji to clear stage 4. Use BB & Maggie to cut Mihawk's health, and Gasparde & Kuma to boost (Kuma won't trigger Mihawk's orb shuffle if he doesn't actually shuffle anything). You can burst Mihawk easily after that.
If you've got the RRs, you can swap Kuma and Gasparde for Squard and Blueno, which offers a more reliable setup with no concerns on orb shuffling. That's the setup I used.
Obviously, if you have Legend Lucci, Hody, or Blackbeard, you're better off building teams around them. I think one of the consistent strategies is to find a way to set your orbs on stage 4, then lock them or clear S4 with a special. That way, you still have your orbs for Mihawk which makes it pretty easily to one-shot him.
Honestly, the Powerhouse quest is extra hard for Global since we lack more JPN-only units there compared to Slasher or Driven. It doesn't help that the orb shuffle nukes characters who would otherwise be MVPs here (Kuma, Hack, frickin' Kyros) and the DOT plus Mihawk's heal makes it hard to outlast him with a tankier team. It's just a rough setup for Global, especially if you're missing characters like 3D2Y Sanji or RR Blueno.
EDIT: In other news, looks like we've got Legend Magellan incoming for JPN. I mentioned a while ago that we were due for another Shooter legend, so this may be that. I'm guessing it'll be an Impel Down batch, with RRs of Hannybal, Domino, Sadie, and a few other wildcards (Saldeath? Bon Clay? Hancock?).