Once a tub gets to the point you describe there is not a lot you can do short of replacement.
Anything else will just be temporary.
You can try silicone, bondo, epoxy. The problem is whatever you use will have nothing behind it for support but a rusty soft spot / hole.
You mother could live longer than you think. I would get 2-3 estimates as to the cost of replacement. At least you will know for sure what the cost would be instead of guessing.
If you have the $$$ to tear down the house you surely have enough to replace a tub for mom.