We carry out Cerakoting, industrially.
The Cerakote process requires the surface to be shot blasted, then ultrasonically cleaned in acetone and baked in an oven to offgas anything remaining, before it is spray painted with Cerakote resin and baked again to cure it. If one misses any step the Cerakote will flake off.
Therefore if you buy a blue barrel, the blue will be removed in the shot blasting process before it is Cerakoted.
The advantage of blued barrels that are then Cerakoted is they are (usually) made from decent steel and are cheaper.
Stainless steels used for gun barrels such as SS410 have a third lower tensile strength than a decent non-stainless steel. In fact old Whitworth fluid pressed steel has a tensile strength of about 630Mpa compared to 450Mpa for SS410, and 230Mpa or so for your regular SS316. Whitworth fluid pressed steel of 1890 has about the same strength as titanium grade 9.
BTW, don't put acetone in your ultrasonic cleaner at home. The vapour cloud can go with a bang if someone approaches with a fag. Especially as we run the cleaners hot. It needs forced ventilation and good seals on the cleaner.
BTW2: I prefer blued barrels and don't Cerakote my own. Easier to fix marks on a blued barrel than on Cerakote.