What reason could u have to increase your Heartbeast skill?

You don't need it to change to your heartbeast form. It is useful only to avoid unwanted transformation (which don´t come as frequently). When you sucumb to twilight, it will determine the Magic Might of the creature but... you won't have a character. Other uses are marginal or require misteries.

What reason could a Bjornaer have to devote xp to that skill instead of the skills of his/her heartbeast or magic theory or any other skill? If only that skill could be used for inner beast initiation or the refinements instead of House Bjornaer Lore... Somebody has houseruled something around this skill?

We have dealt with Bjornaer very little, so I'm rusty on the subject. With a quick scan:

A roll would be required in a hostile Aegis.

I would call for a roll from a Bjornaer in an alien mystic aura, and apply aura penalties (although there may be rules on that).

I would also call for a roll when the magus is under duress; a (bird) mage falling from a great height,* for example, or a mage under attack.

*Yes, it takes a round.

If you check the rules for Theriomorphy and Sensory Magic from HoH:MC, Heartbeast really helps with these virtues.

And many Bjornaer go for the Inner Heartbeast Mystery, don't they? They need a decent skill + Sta to equal 9 to switch to it from the Outer Heartbeast, or they might be "Guys, just give me another round, ok?"

Anyhow, what I wonder is how do Sagas deal with Heartbeast skill botches. "Your boar's snout never goes away" or " Nope, it looks like you are stuck in that form for at least another year" or " You lose 15xp from Heartbeast"

I suppose that it is as any other supernatural skill aligned to magic, so normal penalties apply to, for example, change inside a city. But the thing is that normally a roll is not needed for that.

I have dealt with Heartbeast botches not allowing the magus to change in that scene (stuck in whatever form he was). But later, the player pointed out that he should not have rolled in the first place because it is not needed for a normal change, even in combat.

Edit: BTW, you should never lose xp in a roll in which you cannot win them. Like in any skill roll.

A Stamina + Heartbeast roll of 9 is required to change into an Inner Heartbeast form.

I think OP already had the answer in his first post.
If you are not going to bother with inner mysteries, there is no point bothering with the heartbeast ability.
If you intend to turn mystic in contrast, there will be a point where you need the heartbeast ability. A clever Bjornaer may even invent ... sorry discover ... hitherto unknown mysteries which require the heartbeast ability.

One thing not mentioned is the Personality Trait attached to the animal form being equal to one half the Heartbeast Score.

It could be used as a Sympathy modifier for some refinements and/or become an Essential Virtue in the right circumstances?