Criamon: If you want to be a Criamon, fine. This whole House thing is not a real distinction, but we accept that most magi think it matters. The only true distinctions are between apt and not apt, harmony and strife, freedom and imprisonment. If the tree is good, we'll know by its fruit.
Bjornaer: Everyone has a Heartbeast, and we can usually bring it forth, but we are not going to initiate just anyone. Shapeshifters are not welcome. Merinita and others under Faerie influence are not to be trusted. Magi from Latin Houses have it in for us, so their interest is questionable. There is good reason to keep secret the initiation into the Outer Mystery, even from most of our own House. This must not get into the wrong hands. If our magi seek questionable mysteries, they must be brought back into the fold by any means necessary or be returned to their ancestors.
Verditius: Are you willing to lose the ability to cast formulaic spells without casting tools? Are you willing to be considered a bit "less" by other magi? Can you convince a Verditius magus that it is worth time away from his fabricatorium to initiate you? Then... sure! If the Mysteries of Criamon are self-occluding, the Mysteries of Verditius are self-reinforcing, because of Hubris. You thought you'd just become a great enchanter on top of everything else? Welcome to Hubris. The more magic items you create, the more you need to create them. The more rivals you defeat, the more vendettas you need to have. It's not just a job, it's a way of life. Sure, you can be something else if you want to... but have you built anything cool, lately?
Merinita: The House lacks sufficient organization for generalization to apply. Want to grow closer to the fay to best slaughter them all? There's probably a mystagogue for that, and a whole bunch of faeries utterly delighted to play supporting roles in your story. Illusions? Faerie realms? Systems of correspondence? Botanical mood enhancers? Anything? But of course, you have to be willing to endure the trials your mystagogue to be sets before you, before you even begin to initiate. If you're not committed, you might find yourself charring faeries in exasperation. And can you get enough Merinita, or the right Merinita, to even agree that you belong to their House, FM virtue or not? Just because one Merinita says you qualify, doesn't mean anyone else, in or out of the House will agree.
Currently, the only Mystery group who teaches the HeartBeast is House Bjornaer, and they do it through the Ritual of the Twelve Years. No Familiar is an Ordeal specific to initiation through the Ritual, not an intrinsic property of the Heartbeast. In fact, HoH:MC says something about Bjornaer choosing that Ordeal because she did not care about binding a familiar, and has a story seed about initiating Mundanes -- for which obviously a different script must be used, since they can't have familiars so No Familiar is no Ordeal at all.
So, you could learn the Heartbeast by being taught it as a Gifted student, or through some other initiation. GotF also mentions that it can be learnt as a Path of the Forest. However, I'm sure the Bjornaer would declare Wizard War en masse on any magus with the affrontery to claim their secret without joining the House.
Incidentally, from what we read in HoH:MC, the same seems to apply to Verditius magic. Intheory*, anyone could learn it (or, rather, be initiated into it, since it's not an Ability that you can be taught by virtue of the Gift) without one's formulaic magic being crippled. It's just that Verditius magi all have this spellcasting problem, and they all automatically inflict it on their apprentices when opening their Arts -- or artificially on magi who later join the House to avoid giving them an advantage.
Or maybe he realized that all this stuff about the Enigma and the Axis Mundi was a big joke, and wanted to see if anyone could figure it out on their own!
And for some reason, this title makes me want to find the rest of the joke (punchline doesn't seem right, but maybe it is), because it feels like it has the same beat as: You can tune a piano, but you can't tuna fish.
I know. When I first saw the title I was thinking the question was about MuTe(He) or similar to take a physical house and change it. Maybe "MuTe" could fit into your pun somewhere? Skeletons in a closet not talking... I don't know.