I think plugging the drain, and screwing up the drain flange, and possibly ruining the tub drain seal, are all bad ideas.
Better to spend money having the drains or sewer lateral under the slab snaked as often as needed.
If you try to rely on a plug in the drain, either it will fail, or the sewer line will really fill up and become a solid mess.
In my opinion.
Which is not professional.
But I have encountered almost that exact scenario several times.
A customer who had me install one of those automatic drain plugs once called me in a panic.
It was a self sealing type of floating plug that protected a floor drain in his basement.
He was standing on it, trying to keep a river of **** from oozing out.
I had no suggestions, other than to wait it out, then tear up the basement floor as I suggested months earlier.
His sewer line had sagged like yours, and also had a small blowout from high internal pressure.