'borax' first occurs in 1944, a hundred years after 'borak'.

Well, I had my doubts about this, and subsequently found the following in "Ullmann's Encyclopedia of Industrial Chemistry, 6th edition, Electronic Release, 1999":
The Arabic word for borax, baurach, is found in old manuscripts from ancient Persia and Arabia dating 2000 years ago. ... At the end of the 13th century Marco Polo brought borax from Mongolia to Europe. This became the primary European source for use as a soldering and enameling agent.
Frankly, I believe borak, in the course of time, got assimilated to borax (because the latter is more widely known) rather than the other way round.