It is mentioned in some source material for ArM5 that the canonical reason why Bjornaer magi cannot bind familiars is because the Bjornaer founder decided that having a familiar was not for her and so made the lack of ability to bind a familiar into the ordeal flaw for initiating the Heartbeast virtue. As such it is possible within the rules to e.g. integrate the Heartbeast virtue, rendering magi able to have both Heartbeast and familiar.
I do agree with Oneshot that there are some muddy thematics at play. But personally I dont see any reason why this could not be resolved or even ignored.
Not having a familiar is a serious drawback for magi Bjornaer, and there is no reason why you could not change it if you want to. Personally I think it would only put the Bjornaer on even footing with the Merinita and Criamon in having access to house mysteries and the full spectrum of hermetic magic.
However I would like to note that I dont think the question of "balance" is a valid question to pose in Ars Magica. The game does not have a balance to achieve, and as such any effort to try to balance house Bjornaer is doomed to fail. The game has so many combinations none of which are equally balanced, at least from a mechanical perspective, nor does the game even have a straightforward parameter by which to measure balance again from a mechanical perspective.
The best balance parameter I can come up with is one of stories. In that sense the real question to me is: "Are magi Bjornaer missing out on participating in the story, by giving up the ability to have a familiar in return for having a Heartbeast?". This is a question that each group must answer on their own, but in a sense, they are missing out on the stories that spring from finding and having familiars. They do also gain the stories that come with being both human and animal and if you care to/manage to include it in your stories the much tighter knit community of Bjornaer. However if your game does not include any other Bjornaer to have a community with, then this aspect is much lessened.