My liner is removable, isn't yours? I have a mini metro bag. My tarp liner is completely removable from the inside large area. But not removable from the smaller front pockets or on the lower outside front flap of my bag. Otherwise, I would use adhesive like Gorilla Glue or a fabric glue. I wouldn't use heat near the tarp liner. Can you place something between the 2 layers if you can't seperate them while ironing? Have you google searched Chrome bag images and see if you can find one somebody patched and what they used to do so?
I dunno if you are in the market, but I also got their Bishop wallet and buckle keychain for my keys and the pouch pocket for my sunglasses/cell/garmin, slick system!