Our manual says to add about five gallons of water to empty tank.
You can adjust how full the bowl gets with water after flushing, at least on some models.
There has been a lot of he hawing about bidets on the forums over the last few years but I would either install a whole unit designed or Mhs or the type that fits under the seat.
Believe it or not that spray beside your toilet is marketed under Bidets at Home Depot etc.
It sounds like not flushing long enough or enough water in the flush or a heck of a lot of tp at each flush.
If I understand right the tp in the tank is up near the toilet also?
Why not empty and flush it a bit more often.
Put a Y at the faucet with a valve. use another hose while still connected, use one anyway, and ill drain and flush and add about five gallons back into the tank.