What I read from
elsewhere, TS80 electronics (I'm not talking about heating element now) will die at about 16V. I have not tested, so don't blame me if it dies at lower voltages.
Normally TS80 will use QC3.0 (or QC2.0) protocol to regulated input voltage ie.
maximum power/current to heating element. Also it has software over voltage protection which voltage you can set in menu (maximum setting is 13.0V, I would set it to below 11V). TS80 will shutdown at that voltage, but it won't protect electronics!
this regulator from Aliexpress
That is a constant current driver/regulator. It does
not work with TS80. Constant current = output voltage can be (almost) anything to keep set output current.
You would need constant voltage, a voltage regulator, to output 9V converted from your battery voltage.
Also when playing with batteries some battery protection is good to have: a fuse to limit current in case of shortcircuit and a low voltage protection/alarm to not over discharge battery. Usually Lipo-packs do not have these protections build in.
Have you considered TS100? People use it directly from Lipo batteries. I don't have personal experience of it and I'm not sure what input voltage it accepts. Forum search should find topics about that.