The problem is that you have two distinct conditions.
PID is great to MAINTAIN the temperature, but start-up or such a huge system upset requires a whole different set of parameters for PID.
in this case, PID is not the way to go.
what you have is a thermal mass of predictable value.
you have the ability to add an equal thermal value to that.
what you have already expressed is that you need 100x BTU to heat the space, you add 100x, the space is heated, but you still have 20x that has already been put into the space.
overshoot is ineveitable.
PID would offer that you add some, watch the reaction and then add more...
but thermal is not a process that uses PID. cannot get more simple than that
there is no place in heating that D adds to good control because the process is so slow. I is not all that important.
and the real question is WHERE you measure your process.
at the far side of the room and you will always overshoot. it is inevitable. because there is no way to know how much heat has been applied and is already in the space, before the sensor can detect it. and once the sensor has detected a rise, then the process has already overshot.
if you put the sensor at the heating element, you have no control of the room.
because heat is both convection, radiation and conduction, all at the same time, your space sensor is not going to allow exacting control.
this becomes less and less apparent as you get closer to setpoint, but on start or a huge process change, it is worthless.