Yeah, a factor of two or three over atmospheric is probably enough to get good data for a test like this. (I am an experimental physicist, working with cryogenic systems, including Gifford-McMahon and pulse tube coolers and occasionally Stirling cycle coolers, which are Stirling engines run in reverse). 15 and 30 psi is enough for initial tests, before you decide if it's worth going to higher pressures.
I assume that you won't be running a combustion source inside the pressure vessel - firstly air under pressure will greatly increase the rate of combustion, possible nullifying your experiment, and secondly, at higher partial pressures of oxygen (this includes pressurized air) there's a greater risk of other components of the system igniting. This also applies to a lesser extent to electrical components if there's a risk of sparks.
If you are doing a university type research project, then testing under moderate vacuum would also be good to do, and somewhat safer than high pressure. You should also investigate the local safety regulations about pressure vessels - I don't know about the US, but in the UK, a vessel intended to operate at over 5 bar (75psi) has to be certified, and any modifications have to be certified separately. Cornies don't usually have to, as the design operating pressure is below 5 bar, although they are tested higher.