70-200 f2.8 vs 70-300 f4.5 lens

Ultimately the choice comes down to finances, wishes and weight.
You have not mentioned the first which makes it hard to offer advice. Of course then most of us assume you are looking for cheaper/lighter alternatives. You may not be.

I am going to assume that finances is not much of an issue - then go with the 500mm/f4 and get the 1.4 TC as well. Assuming a full frame camera you are looking at 500-700mm.

For birds being too long just means stepping back. Being too short, means getting close and even with fieldcraft that is not easy.

Don't assume 500 is too long. It really is not.