A strange carpet question

[PinotAndPlaydough]

I should probably start by saying we live in a rented property so our options are limited.
Basically my child has asd, they don’t play games like other children. One of the only thing they like to do is swing on a pull up bar. The problem is they swing so far they are pushing the carpet back when they land and they are pulling up from under the metal door strips.
We have tried putting down a rubber mat and loads of cushions but it’s not working. The carpets and really old and were already lifting but this is a real tripping hazard and looks awful.
I don’t really want to take the bar down because it’s literally all they really like to do. Is there some sort of magic diy trick that I can do to keep the carpets from coming up?

Can you glue the carpet down? Some of ours were glued we found when they were replaced. Might not need to mention it to the landlord ..

Glue the carpet down properly and neatly (might be worth get a handyman in to do this.)
Then buy a rubber backed rug which fits the whole doorway width-wise and then extends as far as possible each way lengthwise.

Don’t know why I never thought of glue!