I have just a couple of things to say.
On the subject of IC opinions. There is nothing wrong with the perception of Pyre as bastards from groups like IRED or PIE. They are faction loyalists who would obviously judge us by our associations. In fact there is really nothing wrong with anyones IC perceptions of us as that is up to each person or corp based on their personal RP. As has been said we are a PMC, formed by Caldari and working under contract with the 24th IC. We employ capsuleers regardless of their RP history or previous loyalties. If your characters hate us or love us, whatever it's all good and all workable as RP.
As for the idea that our very existence destroys or "infects" loyalist FW RP I would have to disagree. For one thing the militias have always had these elements within them. Taking into account "non-roleplayers" one might say that for much of their history ingame the overwhelming majority of any given militia has been people of questionable moral character. This leads one to a sort of quandary. If it seems strange that the Empires allow such pilots to fly under their banner should we not search for an explanation instead of throwing hands up in the air and saying "it's all stupid!"? From my perspective the faction wars are not a big thing, they are brushfire wars on the outskirts between mercenaries, deniable resources and small cores of actual military from the Empires. The outlier is not the PMC like Pyre but the independent capsuleers truly loyal to their faction like PIE, IRED or UK. In my opinion this works quite well with observations of what actually is happening in-game. And when it comes to our very existence somehow polluting that faction I would add that even if you do not agree with my view of the FW, if you add up the players in Pyre, Nauplius' corp and Cult of the Lash I think you are at like all of 60 people maybe. Perhaps that is some significant number of RP'ers when measured against other roleplayers either in our own militia or others but I feel that says more about unrelated problems others are facing than ones we have supposedly created. Especially when you add in the fact that basically no one we have recruited was welcome IC in any of those other organizations anyway.
TLDR: We are not "stealing" "your" potential roleplayers. We are not "infecting" your roleplay. If there is a dearth of people interested in whatever you are doing you should probably come up with something fun to do and then go and recruit some people or something. Good luck and fly dirty.