I would suggest taking a sponge, saturating it with fabric cleaner, and wringing it out really well... Try gently cleaning the lining that way. Once you're done, wipe it dry with a paper towel and leave it open to dry.
On other forums, people have also suggested using baby wipes and Shout spray with a wet rag.
That's the best I can suggest considering it's a fixed liner

Best of luck!