I've never owned the trousse toilette...but it looks like the interior is vuittonite (which is found in the regular and petit buckets from the monogram line).
Unfortunately, peeling vuittonite is a common phenomenon that happens with extensive usage and excessive heat over time. It's normal wear-and-tear.
The only remedy I can think of is to replace the lining of the makeup case. I did that with the petit bucket I used to have...it wasn't expensive.
Hope that helps! Sorry I can't be more optimistic...