I don't see getting access to the improvement lists again being a problem either. I mean, the stat boosts have maximums, right? And it's not like you can advance basic moves twice. So those things will be off the table once they are. but why shouldn't you be able to retire to safety, bring in a second character, or switch playbooks again just because you've already done it once? Is your real objection to the possibility that a character might eventually get all his stats up to +3?
I don´t know, I guess that gaining access againt to a list of improvements that is essentially the same for every playbook strikes me as odd. Not really dangerous or game breaking, except perhaps because the +1 to any stat makes it easier for them to reach +3 on 3 or 4 stats by the second playbook (but anyway they´re already very powerful, so it doesn´t stop them much.).
Maybe the most important thingie about allowing or not the advanced improvements is that, if you get to choose them again, you can reach a 3rd playbook, then change again, etc. If you can´t, your character is nearly finished upon the 2nd playbook. I don´t know which one is correct, and I wouldn´t like to make that decision myself (I´d prefer to see here the "official canon" on the subject).
I guess a brainer would lose his specialty gear, and the gunlugger and battlebabe would lose their custom weapons. Am I right here, or is there anything else they could lose?
In my game, they're gonna lose whatever makes sense. The gunlugger's MG doesn't vanish just because he switched playbooks. "Duke, would you keep your MG and all that shit now that you've gotten this following?" "Oh yeah, it's a gun cult." "Sweet."
If the battlebabe, both character and player, adamantly want to keep their custom sword, there's little you can fictionally do to take it away without resorting to narrative backflips. No, you'd have to actually go into a game with your prep as "Take away the battlebabe's sword". Yeah....
It's an improvement and a big deal, not a start from scratch. They've earned this and gone through hell to get here.
Well, there´s an example where a Hardholder loses his/her holding, and that is indeed a big loss, if we see that hardholder have almost nothing besides a holding. Maybe for playbooks where there´s not much dependance over other people (like the 3 PCs on my group) there´s not much to lose. Anyway, even if the story rules otherwise when we reach that point, I´d like to have some sort of guideline when I reach it (like: ok, you say you don´t want to kill people ever again and become a Hocus, why don´t you give away your custom weapon to better represent it?)