I know a little bit about general barrel care, but more specifically was wondering what to do to a barrel between transferring one beer out and putting another in, taking into account the barrel and styles.

I have a 15 gallon barrel used to age gin, purchased from a local distillery. My first beer in the barrel is a Belgian wit, aging now. Wit uses similar flavoring agents as the gin, so I thought it would be a good first beer in. I expect it will be ready to empty in one to two months. After that, I plan to fill the barrel with gotlandsdricka, a style that I think will work with the then reduced herb character from the gin and wit. I'm hoping to brew the gotlandsdricka soon so I can empty the barrel of the wit and fill immediately so I don't have to rewet/recondition the barrel.

Considering the wit beer and yeast, what would you do to the barrel between the drain and refill? Sulfur stick? Hot rinse (180 F)? Other chemical rinse? Nothing? Any other good recommendations?

This is the chemist in me, but I'd brew a little extra of the second beer and rinse the barrel with it.

Depends on how much yeast is in the barrel from the wit. Since there isnt any brett in there yet I would do a quick rinse with some hot 180f water. Just a gallon or two, all you want to do is get the majority of the yeast out. This has worked for me and kept a barrel going for a year that way. Now I have three barrels two of them sour. The sour barrels I dont worry about excess yeast. Dead yeast is food for the bugs.