Just to wrap it up, to remove 比我, I think the following should be correct:
他快高点儿了 (she`ll soon become taller)
他快要跑得快点儿了 (she`ll soon be able to run faster (she trains a lot))
Both sounds unatural and ambiguous.
他快高点儿了:
1. No verb. Usually we use 长高 to say "grow taller".
2. 高点儿 is too colloquial.
3. Why 高点儿 (taller a little bit)? There's no standard.
4. 快... means "soon, about to", but it usually take a long time for a person to become taller. We can't wait for it.
You can say: 他快长高了 or 他很快会变高的 or 他很快就会长得更高 etc.
他快要跑得快点儿了:
1. 快点儿 is usually a colloquial suggestion to ask someone to "get faster!"
2. Two 快 with different functions in one sentence is confusing.
You can say: 他快要加速了 "He's going to speed up", 他马上就会跑得更快了 etc.
I don't advise you to use [快+verb+得+adjective].
得 expresses a feeling of "result", while 快 means the thing has not happened yet. They don't match. I think this is why [快+verb+得+adjective] sounds unnatural.