But with this change they would have access to familiars, until they no longer do.
For a Bjornaer magi there would be no past where they did not have access to a familiar.
(Canonically, the inability to bind a familiar counts as a Minor Ordeal when going through the Ritual of Twelve Years.)
Like I said in my comments, it is mostly there for flavour.
Playable. I don't think it is imbalancing, but that is no reason to adopt it.
If the saga takes a form where Bjornar is disadvantaged, I think it is ok not to play Bjornær. That makes the house rule superfluous.
House rules in general, I find to be an extra burden, and they should have a very strong justification to accept. This is not even close.
The heartbeast form with an animal companion is, I think, a little bit odd. A minor point, maybe.
In short. Playable, but I would rather not.
I would totally allow it, mostly because when I think of Heartbeast I think of Animagi from Harry Potter which leads me to thinking of three boys becoming such for their forth friend and then I also like to imagine Harry himself learning the ability, though that is not canon.
Anyway, to me, there is no intrinsic reason why not and because of that I say go for it.
Imagine the impact to the order if someone managed to integrate the heartbeast mystery...
I have imagined that, and my imagination of it is that it has surprisingly few consequences.
The most interesting effect, to me, is that it would make it so that all magi have a lot less in common with ordinary folk and likewise a lot more in common with each other, as integration of the heartbeast would mean that all magi are simultaneously both man and beast, truly (Essentially) rather than being people who can learn to wear the skin of beasts as one ability among many.
It would also mean the bjorner house loses its distinctiveness, being a mystery cult with no primary mystery, and yet as long as they take apprentices they will not simply vanish.
That would make them no different than most of the other houses. Most of the houses have no unique magic powers, what distinguishes them from each other is the attitudes of their members.
Your views on the familiar are a bit idealized I think.
For my sagas, familiars are individuals. Most of the time, those individuals are not interested in lab work. To get help, the maga may have to bargain. She probably can count on the familiar for longevity rituals and empowering the bond, but probably not much. It will be difficult to ask them to study magic theory books alone for a season (Ha! Whatever!). However, I agree it may depend on the familiar.
Moreover, familiars have drawbacks. One can have arguments with her familiar. The familiar may get targeted or kidnapped. Or even get killed. From time to time, it's a source of problem. Perhaps your familiar will cheat (and stole vis for his consumption), or he will fight other familiars, or even magi. Perhaps, he will travel alone from time to time. Perhaps he don't like when you help those petty humans and will try to convince you to act as he prefer?
My view is heavily influenced by the 4th edition, I think. If you don't see the familiar only as a tool, then suddenly it's not overpowered anymore... unless your familiar is called Sordus, of course. 
TBH. I think you hold the idealised view, and the opposing view is simply degenerate ... corrupted. 
Something to keep in mind here- RAW, nothing prevents a bjorner from having an animal friend- indeed it would be very much in character for them to have one. A wolf bjorner might have a pack mate, a raven, or so forth. A magical animal friend who learns magic theory can still serve as a lab assistant, and do many of the things a familiar does, the major difference is that it cannot have the three cords.
Again given the bjorner history regarding the binding of spirits, it seems to me there is a very strong cultural reason for them to not have a familiar bound by the three cords.
I don't see anything in the books that suggest Bjornaer magi would have a cultural problem with familiars.
The witch-cult that Birna came from supposedly enslaved spirits and could then take their shape, and this is why House Bjornaer are very suspicious of shapeshifters. While it would not be unreasonable to extend that suspicion to any binding of spirits, I can't find anything to suggest this is actually the case.
Case in point: Birna was close friends with Merinita, even though Merinita was the one who introduced the binding of familiars to the Order. I really doubt they would have been close friends if Birna had thought there was something wrong about binding a familiar.
Keep in mind that a familiar must agree to be bound - it is a voluntary bond which works both ways, not enslavement or servitude.
I must admit, I thought not having the cords is the big deal. I didn't realise in Monty Hall Ars Majica a familiar is also a lab assistant.
I'm a fan of Laurent's interpretation. A familiar is a living being with it's own goals and desire. The familiar probably doesn't have a desire to spend 2 seasons studying with the companion with good teacher puiisant and affinity in teaching, communication 5, magic theory 6, etc.
If a group is going the full monty hall, does game balance even matter at that point?
In reverse order
Familiars do tend to share certain personality traits with their mage though.
And I agree, much as I love playing in the lab, I too tend to think this can get out of hand very quickly.
That's not Monty Hall, that's straight out of core.
It's a pretty fundamental part of why everyone's been talking up the power and usefulness of familiars for years. Maybe have a look at Anulis Connectens?
The Cords are awesome. Having someone you can trust, effectively allowing you to act in multiple places at the same time can be useful. An assistant in the lab can be very useful, especially one that does not count against the number of assistant that you can normally have.
But to me, all of that pales next to the ability to invest the Bond with effects that do not cause Warping!
That's true. Empowering the bond and the chords are great. Especially long term.
However, the heartbeast has a few advantages too. And you don't have to enchant or reach tremendous lab total to get them. Even apprentices already initiated can pretend to:
In fact, most of the time, I find young Bjornaer magi much more powerful than their sodales.
This depends a bit on the choice of Heartbeast and the focus/theme(s) and length of the saga.
But I'm not disagreeing.
yes, the familiar is a character, not having the bonds doesn't change that. My point here is that the bonds are the only aspect of a familiar which are missing for a bjorner.
Also there is space between "forging bonds is evil and I must oppose the magus who introduced the idea" and "I'm not comfortable doing this given my background of running away from people who enslaved animal spirits to change shapes"
It would be like coming out of the pre-civil war south and recognizing that BDSM is a valid lifestyle choice but not being comfortable being a dom because you have seen real slavery.
I am not sure BDSM leaves quite the same scar on the soul as the magical cords tying you the magus to the familiar for all eternity. YRMV of course. That´s R for relationship.
Whether the bonds leave a scar on the soul or not is also debatable. In either case past experience can makes something personally unacceptable even when you realize it is fine for other people.
I see your point, but
a) Apart from Birna herself few if any Bjornaer magi will have any personal experience with the enslavement of animal spirits
b) The binding of a familiar is a bonding of willing equals, not that of master and servant (let alone master and slave.)
The way it is described in canon, the bond certainly leaves a scar if it is broken, so you can choose between scarring or irrevokable ...