I haven't manually peeled my lips in about a month or so and almost always just let the saline solution and water afterwards "take care of" the build up. Although I do have build up on my lips most of the time and it's noticable for people, it's not like they look disgusting or anything. They usually just look white to other people, like I have some kind of sunscreen on them or so. It's hard, but I go to work every day. Talking is often the hardest part, because it's pretty uncomfortable at times, but not that much more then when I was manually peeling.
I do sometimes feel like I'm using too much water or saline solution and that that way I'm washing off too much build up too early, but it's probably way better then peeling them manually.
You should really consider doing the same. Why risk having this any longer than you have to just to have a few weeks or months of not having a lot of build up on your lips that isn't really all that bad anyway.