They were gone the second another city dangled a brand-new facility at them, no matter what their name is. The former Zephyr Field needs a lot of work to modernize and right-size it.
Completely anecdotal, but I saw way more Baby Cakes merch on the streets in New Orleans and other areas than I ever saw with the Zephyrs (and I say that as someone who's had Zephyrs gear ever since they came to town!) Even I own the NO hat with the king cake and one with the baby smiling wearing beads - they are great hats at Carnival time. Never was a fan of the "main" logo though with the angry Tommy Pickles as you put it.
I always thought the plan floated around was the current ownership team would buy and relocate a Southern League AA team, which would mean the Baby Cakes moniker would live on at the lower level. I wonder if that's still the case, because I'm not sure which team would be ripe for the picking and I also think that would be dependent on remodeling the stadium - it's too big for AAA baseball now, it's downright cavernous for a AA team.
As for whether Benson gets involved, I'm not sure I'd want that. The minor league team would be even more of an afterthought in the facility than the Pelicans are now.