Tell them to stfu. I-RED and CAIN have been the target for much flak recently, mostly done by ignorant characters that can't even imagine what these entities have done way before FW was in place. Those entities have accomplished million times what most nowadays "loyalists" have accomplished in all their pilot lives in FW (which is, may I remind you, a stupid nonsensical mechanism, IC wise). CAIN have held for the state entire constellations in nullsec. CVA has lived for so many years holding providence. I-RED has done a bit of all this too, either as allies or by itself. Nobody would shit EM for that people just know they are Matari loyalists, and that better not to fuck with them. It would be like asking CVA how the hell they could be Amarr loyalists when they have not even touched FW. That's properly ludicrous, you just have all the argument in your favor here really.
It has always been the case between FW supporters and old RP alliances anyway. It was the same between EM and the TLF, it was the same between CVA and the 24IC when I still was in KotMC, and it was already the same between I-RED and the Caldari Militia either. Most of the time, it was a matter of conflict between NRDS stances of the former alliances against all the rogue pirate elements of the militias, thus creating a lot of "alliance loyalists vs militia of the same side" incidents in space. Especially since at that time alliances could not even enter FW.
As for the Caldari Prime titan, let them talk. Most characters will try anything to make their black and white tainted glasses work everytime and spread lies or whatever to paint people as race traitors... I would say, it's expected now from the Eve RP milieu, the same way the Eve universe expects you to deal sooner or later with OOC betrayal, drama, ruthless and uncouth actions, etc. All of this also exists ICly and it has to be experienced and dealt with eventually.
I have faced the same thing in my time in KotMC against PIE Inc for playing a character different from the mainstream (and still occasionally do, but now I have faced it with success it has just turned into mutual indifference or just some kind of cold war). Somehow I would even say that's maybe not such a bad thing for Kat to have that false reputation, where all her friends probably know better. Can be interesting ICly to play with. I know I have a lot of fun at times to play the same thing myself. Sometimes you face successes, sometimes you meet public humiliation...
Also, as Nico pointed out, you can also start your own loyalist wing. If it succeeds like KotMC did for us, you will be able to rival everything. At least, at its golden age, KotMC rivaled PIE Inc and was the main Amarrian RP vector above everyone else that was basically able to tell even the older corps to stfu considering its achievements and popularity. Granted, it was a FW corp doing stuff in FW... But it was a liberal oriented corp, and those usually have the big advantage to draw a lot of friends, even in foreign factions or independent characters. It gives a lot more support than usually expected.
So your advice is join them or fight them? There's no working with the other side?
It's tricky but you can. Question their authority on the matter. Who the hell are they ? Independant capsuleers ? Not even military capsuleers ? What a joke of loyalists, really.
Of course, all i'm offering is conflict, since it's the only thing I have done in the same case in all my eve career... The trick is just to show that your side is not a fickle one, with solid backings and arguments behind, and that it has to be reckoned as such, and serves the same interests, just in a different way.
I have unfortunately no precise idea of events to do. But you can still organize a lot of parties axed around IC caldari fluff, like duty ceremonies (so say we all!), like Malkalen you already do, megacorp meetings or press conferences with foreigners, whatever. Attract people to your own caldariness and spread that caldariness everywhere. I can guarantee you that if that gets active enough, people will have to reckon your character as a pillar of Caldari fluff. You already do that with Malkalen, or your own background corp for your character. Include people in these, or in new ones. See the PIE ball ? It's all about prestige. Create your own prestige.
Even better, start to show your character at most important events. Represent Caldari instead of others. Steal them the occasion to do so every time. Kat should show herself more. Orange is trying to organize something on the Caldari mindset on another thread here. Go there and make an impression. Stomp her feet on the faces of her enemies in the process. Reap the profit.
You don't have to do lip service to the patriot bloc if your own bloc is influential enough. Your recent attacks on the patriots on the IGS were awesome. That's what I have been expecting to see again after FW, which is sub factional inner conflict. Politics. Fuck the unity.