Is there any possibility of unlinking Brawl from the Trials-War schedule? Including Brawl makes it take that much longer to get to the next War/Trials, and considering that each team has no thematic connections to its element I'm not even sure why Masters of elements become Bosses in the first place.
Not if you want anyone at all to play in brawl, no.
Typically the last few rounds of war involve not very many participants, so that would be an excellent time to start preparations, applications, etc. so that brawl can start almost immediately after war