There's no real legitimate way, no decision to "reform the republic" or anything like that. I THINK if you're inherited by a MR of your dynasty, that'll make you a MR again? Not really sure. But regardless, it's not exactly a legit way. In my view, at least.
I wouldn't mind it [as much, it'd still be a bit annoying] if the game didn't shove you to Feudal at the drop of a hat. Revolt? Feudal. Lose your capital? Feudal. Declare war to take back the republic? Feudal. That last one I can *kinda* see, the idea that you're destroying the whole "republic" part by warring over the title. But a) Venice (i.e. the Ur Merchant Republic) and especially Genoa had plenty of wars over the title, and they didn't suddenly become a King. b) The game doesn't TELL you that it'll become feudal if you war over the title. And c) I still hold mostly cities, and all my counties are city-capitaled (only some Baronys in the capital of the republic for troops) so that's really annoying, suddenly I'm weak as shit because I hold "wrong holdings" because suddenly feudal.
Honestly, republics should be overhauled.
The patricians should be unhooked from the title of the republic and become their own "government" which is linked to a republic. With the added benefit that a house of merchants that gets fucked over can flee to a different republic, because they're not intrinsic TO that republic, they're free-form; They could even seek refuge in a noblemans hall, where they give technology and cash from their trading posts in exchange for refuge! And they can retain their trading posts when they flee, with a bonus to retention chance if they have garrisons in the posts (Hey, it gives you a reason beyond some small bonus to retinue to build garrisons that way ;P)
Add in a more floating number of merchant houses instead of the "you get five" thing we have now, with a total limit of like. 10? After which the republic is too full to add any more. Add in a *bit* more infighting and a natural level of "we don't want more houses" happens, because the cost of fighting is worse than the bonus of more trade posts. That way, houses can jump between them as they get pushed out and you could get random events for a new house to form as some merchant gets cash and can build a dynastic house.
Make the technologies apply to the house instead of the province, so a merchant republic can "bring knowledge to Europe" if they get exiled (read: not suddenly have a tech score of 1 in Trading and can't actually DO anything any more) with a bonus to tech in the province they're exiled to. Plus, you can have scheming to acquire plans and documents between houses if they all have their own tech level! This is already sorta possible; Nomadic rulers have their tech apply to their capital camp, not to any specific province, so it's not unworkable.
And finally: Republican Tradition. Bring that in, from EU3/4. The more wars are fought by the patricians for the title, the lower tradition. The more a single house is elected, the lower the tradition. Add in a "Republican Dictatorship" transitional government, which is identical to a Merchant Republic except that upon death of the ruler, unless it's to a different merchant house, it becomes a feudal monarchy or iqta or whatever it should be. THAT way you get the whole "You're not REALLY a republic any more, are you?" without it feeling or BEING a bug, like happens now.
Maybe every ruler gets a tab like the MR tab, and for nonMR rulers, it lists the merchant houses (in the world? or just in their realm?) with a section specifically for those in your court. (If just in your realm, then a ledger page for the whole worlds merchant houses would be in order)
I feel like all of that would make playing a merchant house a lot more fun and less excruciating. Especially if they fix the bugs too.
;P Plus, it feels more natural this way; You don't just suddenly "oh hey here's 4 random houses" when you create a merchant republic, they're already extant and generated by random events in the world as rich merchants become dynastic and work for their noble lords, seeking a republic to work and live in and acquire power for themselves. And it makes interaction with a republic as a noble lord more fun, instead of just "make republic -> get money."
(and a bit more ideas I kinda forgot while typing as new ideas kept popping up, this is really off the cuff so feel free to poke holes in it)