I understand your setup as follows (in flow direction):
* compressor with cutout set to 150 PSI
* tank (T1) rated for compressor, with safety valve
* (planned) pressure regulator (PR)
* additional tank (T2), rated 120 PSI, with safety valve
If this is so the safety valve of the second tank will protect you unless you tinkered with or replaced this safety valve.
Now, for the planned pressure regulator, check the flow rate you need and what the compressor delivers. Find the pressure differential ($\Delta p$, I'd expect 0.5 bar / 7.3 PSI) of the PR at this flow rate. 120 PSI + $\Delta p$ should be below the cutout point of the compressor, or you wont fill your T2.
At startup, the compressor will fill T1, at ~127 PSI (or whatever depending on the PR) T2 will fill until 120 PSI are reached, then PR will close, pressure in T1 reaches cutout point.
Depending on the cutin point of the compressor and relative sizes of T1 and T2 you may get frequent on/off cycles. PR is an additional pressure loss in your system.
The ultimate problem you want to solve appears to be frequent, noisy runtime of the motor. 568
An additional tank won't change the total runtime for a given air demand much, at best it will make longer uptimes and longer downtimes (a good thing in itself IMO).
If someone tinkers with the PR, the worst I see happening is the safety valve venting provided no one tinkered with the safety valve and inefficient operation of the compressor.
However I'd advise a preset pressure regulator.
Thoroughly check if the setup I assume is indeed what you have. Wait with building the thing until the hive mind here had the time to check my reasoning and find possible flaws.
In the long run, a compressor station sized for your application may be more economical.
ETA: The cutin point for the compressor is stated to be at 120 PSI, with the PR set to 120 PSI it is unlikely the pressure in Tank 1 will fall to 120PSI after the initial filling. See if your tools can operate at 100 PSI: 100 PSI in T2 + loss in PR = approx. 107 PSI + plus some safety to account for inaccuracies should be below the cutin point (if that can't be adjusted).
Let me restate that you should not, in any way, fiddle with the safety valves. Safety aside, for the setup to work you need a lower pressure in the second tank. 35