Your daughter is adorable.
I think you have hit upon something that happens a lot, even here in the US. The same thing happened to us about 8 years ago. In many cases, if you leave the hemangioma alone, it will often just resolve itself over the course of a few years. My daughter's hand/wrist hemangioma went away totally on its own. But lips do tend to be a little more at risk for issues. My other daughter has a lip hemangioma and she did need oral steroids as well as surgery and laser. But hers was also a lot bigger than what your daughter's looks like right now.
So anyway, perhaps you could keep taking pictures on a weekly basis and keep a close eye out for problems/ulcerations. It seems like your doctor may not be aware of some of the newer treatment options available these days..so watch it closely and then maybe you can ask for a referral at the first sign of a problem. Your pictures can help to document any changes. I'm not sure if you are able to get a second opinion, but maybe that can be an option too?
If you look above on the link that says "Resources" you can go to "Find a Doctor " and it lists some doctors in England that might be a resource for you, although I'm not sure if they are close-by where you live.
Good luck! Stay in touch with how it goes!