Think of it as a verb which requires to specify who is to be excused.
The most common use is that one asks excusing on their own behalf: "entschuldigen Sie mich" = "excuse me". In this case, you're asking that somebody asks that you be excused. So, "entschuldigen Sie mich" means that I should be excused. And even when it's another person doing the asking, Mr. Krause should excuse me, and not somebody else, certainly not the asking person.
"Bitte entschuldigen Sie sich für mich bei Herrn Krause." sounds grammatically correct, but with a very strange meaning. If we have three people, A, B and Krause, and A says this sentence to B, it would mean that:
- A finds that B has done some kind of transgression against Mr. Krause
- A requests that B goes to Herr Krause and asks Herr Krause that they (B) be excused/exculpated
- for some unfathomable reason, the whole excuse is to be done for A's benefit. It's as if B did something that they (B) don't think they should ask an excuse about, but A is so mortified about it, that they are asking B to go ask for an excuse as a favor to them.