I assume you mean you just lose access to the book on your phone’s app but not your Kindle?
I have the apps on both my phone and my IPad but I only read books on my Kindle, never on my phone or iPad. I download the books onto my Kindle as soon as they are available and then turn my Kindle on airplane mode. I can then read the books as long as I want if I keep it in airplane mode even tho I have the apps on my other devices.
I do listen to a lot of audiobooks on my phone so unfortunately I do have to finish those within the Libby due date.
I mentioned the airplane mode trick last night to my book club and was surprised no one who uses Kindles knew to turn airplane mode on to save battery as well as keep the book downloaded until they finish. They were happy to learn that trick.