Hello slovac
It's an interesting question. I'm sure that we've discovered differences between AmE and BrE on similar questions in the past, though I can't put my finger on them at present. From memory, BrE was more inclined to use the whole set of auxiliary verbs than AmE was.
Your current example seems to me to raise a slightly different issue.
Personally, I feel rather uncomfortable with both
He did what he should do/He did what he should. However, I wouldn't feel uncomfortable with any of:
He did what he felt he should do;
He did what he felt he should;
He did what he had to;
He did what he had to do.
That said, I'm happy with both
He did what he should have and
He did what he should have done - though I'm more likely to say the second.
I'm not sure I can explain any of this

.