The Pirate factions are way, way underneath the 'big four' in terms of territory, forces, and ability to fight the empires in any meaningful sense. They are nuisances at most, not existential threats. The Blood Raiders are never going to attack Kador and take a high sec system; they would be summarily pimp-slapped back into null, even if they did some all-out ultimate gamble attack and managed to flip the system temporarily.
As stated above their 'sovereignty' is way out in the middle of nowhere (development, resource, and population wise), so owning 15 systems of Angel or Serpentis space in null is probably the same as a single system in lowsec or a 1/4 of a system in high sec for the good it will do you 'empire' wise.
You can reference any number of the news articles involving the empires going after the pirate factions militarily, generally kicking their butts pretty handily with only the commitment of a relatively minor assault fleet. For example the Amarr 7th Fleet by themselves completely wiping the Raiders out of the Bleak Lands in a series of articles a few years back.
TLDR none of the pirate factions can directly challenge the 'big four' in any meaningful sense, aside from constant raiding and stretching of their resources. The pirate factions are likely capable of a direct assault with limited gains, only to have their asses kicked immediately afterwords if they stuck around.
Minor quibble:
Khanid Kingdom eats pirate factions for lunch, or at least at a similar rate as the big four. Don't forget they fought a huge civil war with the Empire proper and managed to keep them at the borders. They say that might be more due to the new Emperor after Khanid took off needing to consolidate forces, but the Empire did try and invade the Kingdom on several subsequent occasions and they were beaten back, along with about 400 years of a not so cold 'cold war' along the border between the two factions until the recent thaw (after Jamyl). A lot of the old Khanid info dumps had to do with them needing to trade with the Caldari for high tech military gear (mostly missiles and shielding) to try and one-up the Amarr on their border since they didn't have nearly the Amarr Navy numbers, hence the need for all the 'high tech' T2 ships to combat the Imperial Navy lolblobs.
EDIT: As most of Kingdom is in lowsec I would ammend that they likely have a much higher level of pirate 'harassment' than high sec Empire. At least that's the way I've spun a lot of my lore being a Kingdom RPer. IE if you join the Royal Khanid Navy as a baseliner you are probably definitely going to see action, and probably against Blood Raiders. If you join the Imperial Navy you could end up in a million different postings against a huge number of foes.