Glue the liner in place while folded to near the closed position. Then there is not so much to bunch up, the flap wants to close on its own, and when opening all the way, the liner has to stretch some. Results in a permanent curve in the flap.
If you use really thin leather for the liner, the life of the purse will be reduced due to accidents and abrasion putting things in and taking them out. I used to use what was called skiver leather that is paper thin. It's okay for wallet liners when you have a hidden pocket for bills. Never used it on purses.
Cut the tails several inches long and back stitch using saddle stitch method manually. If you use leather point needles, they shear the thread when back stitching. So nicer, cleaner job finishing the ends off manually.