Why is "bream" pronounced as "brim"?

Every time I catch an Acanthopagrus australis, commonly known as a yellowfin bream, I wonder why its name is prounced "brim", (as in the same way you would pronounce the brim of a hat).

Merriam-Webster does list the secondary pronunciation, /brēm/. However, when I first uttered the word in that manner, I was immediately admonished by at least three people who were within earshot, and quickly learnt that fishermen (or at the very least, Australian ones,) always say /brɪm/.

Why is this the case? Does it have something to do with the way in which the name was derived?

It appears that it used to be pronounced brim in parts of the U.K. as well. From The Edinburgh encyclopedia (1830), we have

On the coasts of Barbary is found the greatest abundance of excellent fish, particularly mullet, brim, anchovies, sardines, herring, mackarel, cod, skaite, soles, plaice, turbot, turtle.

So presumably, both pronunciations existed in the U.K. at one point, but bream has become the accepted pronunciation in the U.K., while brim is the accepted one in Australia.

In Britain, bream rhymes with dream, so part of your question "why is bream pronounced as it is [in Australia]" generalises to the question "how did Australian accents come to differ from British accents?". This is an interesting bit on the Australian vowel shift and how it started to have an affect within just a few generations: http://clas.mq.edu.au/voices/history-accent-change .