Is there any writen reference to that word? I've been looking to put it in futhark, but apparently there is a debate about its origin(bear skin or shirtless). So, how would they write it down(younger futhark I guess)? Is there any reference to the word in text from that period?
I'm definitely not a translator, but if you're just looking for the places where the word occurs in translations, the Henry Adams Bellows translation of the Poetic Edda uses it several times (you can read it in pdf here, with footnotes on translation). It also occurs in the Arthur Gilchrist Brodeur translation of Gylfaginning on page 73 and multiple times on page 72 of his translation of Skaldskaparmal.
If you want an easily searchable page of Norse texts(or other mythologies, for that matter), sacred-texts is a good place to start, which is where I got all these references, though from what I've seen it only has older translations which has its own pitfalls. If nothing else it should give you a good idea of where to look for comparisons.
The word appears in various sagas, including Yngling's saga and Egill Skallagrímsson's saga and are often not very nice people.
There is, indeed, a debate on whether the word originally comes from a root ber- (hence old norse bjǫrn, cognate with english "bear") or berr (cognate with english "bare"), however, either way, the expected runic form (younger futhark, as you said correctly), would be:
It is impossible that this confusion happened even among the old norse people and different interpretations were kept by different people, making the confusion even bigger.