Hi there! My daughter has a lip hemangioma as well. Since she is 8 years old, the treatments were different back then, so I don't have any experience with propanolol. With that being said, personally, I would see a vascular anomaly specialist or pediatric dermatologist first before entertaining the idea of propanolol. Just because they deal with these things all the time and may be able to give you a different perspective from a pediatrician. It sounds promising that it hasn't been growing rapidly lately so that might be something they would consider. Also, they will have experience with lip hemangiomas which can be tricky at times. My daughter did have treatment (oral steroids) because her lip ulcerated. It was worthwhile to us for that reason and it helped to slow the growth and preserve her lip line which was important to us. My other daughter had a really big hemangioma on her hand and wrist and our specialist encouraged us to wait it out and much to my surprise it totally went away on its own. I wouldn't have believed it if I hadn't seen it myself! The best medicine for her (for me! haha! ) was patience. By the time she was 1 it was barely noticeable. So like I said, going to a specialist might be helpful to you in exploring your options. I will attach my 2 daughters' photo albums just as a point of reference for you; so you can see how my daughter's lip progressed with treatment and then also how my other daughter's involuted without treatment.
Best of luck! Hope this helps! Hopefully others will chime in about propanolol.