I wonder what is the current absorption on batteries due the fact they become warm.
The fact the battery are getting hot means that there's an high power draw. Now it's true that NiMh and NiCd have a lower internal resistance, so they could deliver more power, especially peak power, but the lower nominal voltage and higher current could cause problems.
I am a radio amateur and have some portable rigs designed to run at nominal 12 V/13,8V to be used eventually on a car. The old FT290 Yaesu has NINE cells on the battery holder:
https://www.flickr.com/photos/2w0daa/14628273554
The newer FT-817 uses 8 AA cells instead of D cells and they could be either Alkaline and NiMh, but the holder is different and there's a sense pin used to disable recharging of alkalines.
https://www.lxrobotics.com/wp-content/uploads/FT-817ND-YF-122S-01.jpg
The HT Vx-6 has a rechargeable lithium battery or an alkaline adapter, if you put NiMh the output voltage is too low and the HT does not work.
I think that on the Privia the battery compartment was an afterthought so the power supply unit could misbehave if the voltage is under 12 V. If the original power supply is regulated this could be a sign that the DC supply has to be clean.
Anyway recheargeable alkalines and regular alkalines are almost the same, if you like fireworks, you could try to recharge regular ones. The biggest problem is that is missing the vent for making the excess hydrogen to exit.