>>> Would something like adding small amounts of the fat liquor carefully to the sweat help stabilize it?
Yes and no, depending on how contaminated the leather sweatband is.
Fatliquor-5.0 has a pH value of 5 and has a negative (-ve) ionic charge.
Leather is an amphoteric material and has to be pH balanced back to a positive (+ve) ionic charge before a hydrogen bond can take place.
It is this hydrogen bonding that imparts suppleness and tensile strength to the leather.