I feel like 頑張ってる has two meanings. The first meaning is, as you said, "I'm doing my best" and the second is "I'm trying to make it". For example, you can say「宿題頑張るよ」in the sense of "I'll finish my homework." I think the latter meaning is more common in many cases.
Now, let us consider「今、頑張ってるから、なんとかするよ」. I feel the latter more and 頑張ってる becomes almost identical to なんとかする in this context. It's a bit awkward to connect the same passages with a causal relationship, right? So to speak, it's like saying "I will make it because I'm trying to make it". This is the reason why you can't use なんとかするよ.
Furthermore, it's possible that the speaker is cheering up someone else.
In this case, なんとかするよ is more unnatural and なんとかなるよ becomes the only option.
UPDATE: なんとかなるよ has a nuance of "It's going to be alright, mate" and it's a term used to be optimistic. なんとかする doesn't have this meaning at all and is therefore inappropriate.