Name Withheld
The real problem with improvising a tub to accommodate more water, as you suggest, has less to do with potential leakage than with the poor structural support. When you fill a tub, water has a lot of weight pressing in all directions. A tub has enough support to contain this pressure. But if you were to use the walls for this support, they would not be strong enough. While the walls would not come crashing down, you would place undue stress on the walls.
If you want to go the tile route, you might be able to have a carpenter construct some bracing and then a tiler lay tile over that.
Or, remove the existing tub and replace with a larger-volume tub.
In either case, you also need to consider whether you need additional bracing under the tub to support the additional weight.
It's just not something I could advise.
Since you say your housing situation is temporary, it sounds almost like a rental and that construction of the magnitude I mention above might not even be possible.

