I am not a commander but was fairly shocked by this video of a Wizz crew's decision making. This is not an operation's decision either in my view, this is a command decision.
I do not know the Wizzair company SOPs but it is fairly common , if not the norm nowadays, for a crew to counsult their OPS or Tech to make that decision. If the unreliable airpseed was due to a bird part obstucting the pitot, it often clear itself with the speed, and if indications go back to normal, I do not see a problem with the Captain decsion here..
As to ATC asking a question durin a PAN-PAN, it is fairly standard. PAN is not a emergency and does not require any special handling. The stand by , yes. not the PAN
Finally as far as I know EIN is still a military airbase , so ATC is military. , but things might have changed since the last tiem I flew there. so waiting to be corrected if it changed. Just to say that military controllers are fairly used to PAN calls and well aware of the differences betwenn a PAN-PAN and a MAYDAY.