Hi Mark,
Wow. Adding mass should theoretically reduce gains so you would increase overall gain to compensate. Increase PIDs proportionally. But then if the system has a resonance such as mass-spring-mass such as motor-leadscrew-table then the resonance would shift lower and might cause problems. If the mass isn’t rigidly mounted to table that would add a new resonance.
Any amplifier saturation could also cause problems.
Tuning with a Bode plot, without the extra mass, we look at the 0db Mag crossover frequency and check if there is good phase margin. But then we imagine what if we added mass which would shift the Magnitude down and it’s crossover to a lower frequency would there still be good phase margin?
HTH