I did not read all the responses. But, from your original post, here is my advice.
You should not be taking more than 3 shots at a time without letting the barrel completely cool anyway.
Absolutely work up your load with the suppressor, if that is how you will be shooting it.
If you are shooting a ladder test, start at your lowest charge weight, shot first 3, let cool 30 minutes. Shoot 4-6, let cool 30 minutes. Shoot 7-9, let cool 30 minutes.
If you are doing a OCW test, same practice.
I usually load up 5 to 6 three round groups for both charge weight (week 1) and seating depth (week 2).
Shoot first 3, let cool 30 minutes.
Shoot next 3, let cool 30 minutes.
Etc.
Take more than 1 gun to shoot during cooling cycles. I take 4 to 5 each week. And plan to spend 3-5 hours at the range during load development.