T/P valves open on high pressure or high temperature. A softener can not increase temperature but, if you have higher water pressure during the night than during the day, as most city water users do, and have a control valve that can cause water hammer as it goes from one cycle position to another during regeneration, that can quadruple your main line water pressure and slightly for a fraction of a second open the T/P. Many used T/Ps usually do not reseal when opened but, the softener also reduces water pressure during regeneration because it allows water to go to drain.
For <$15 you can buy a recording water pressure gauge and put it on a utility sink faucet or outside faucet, turn the faucet on and leave it overnight and record the max pressure in your house. If it is over say 75 psi, you need a pressure regulator valve or to adjust or replace your old one.
An expansion tank for the heater will accept out flow due to high temperature expansion of the volume of water filling the tank. It will not reduce pressure or temp so I'm not sure it will solve the problem without knowing the cause but it might treat the symptom.
If you are on your own well, you have an expansion tank; your pressure tank.
I suspect you already have but I'll mention this in case you haven't. You should check the floor around the heater every morning to make sure the leak is only on the night the softener regenerates because it would be very unlikely the softener is the cause of the T/P leaking.
Softeners should be plumbed with 7-10' of tubing from their outlet to the inlet of a water heater. That's so during regeneration warm/hot water can't back feed out of the heater and get into the softener where it can damage the distributor tube allowing resin beads out of the softener into the water lines and fixtures.