The word "Brexit" is a noun, meaning "the exit of Britain from the EU".
The noun "exit" is exitus, fourth declension.
Therefore a natural analogue of the English "Brexit" would be Brexitus.
There might be some use for a verb brexire, but I believe the noun is most relevant.
I have seen "Brexit" used a number of times, and in all cases it has been a noun — read and write complete sentences about Brexit to see how it behaves.1
I think it is very important that the word is recognized well, so I urge you to use some form of Br[itannia] and exire/exitus.
If you use a different word, you are better off explaining the whole thing.
However, my impression is that such portmanteaus would not be idiomatic third century Latin, so you would need to explain (i.e., say "the British exit" or something) instead of coining a new word (like "Brexit").
I think words like Brabitus or Bregressio are way too hard to understand, and are therefore not that good choices for ordinary communication.
I can't keep you from saying Brexodus, but I will struggle to understand if you do so (outside this question where I know to expect things like that).
Nuntii Latini had a news item on Brexit, but it makes no use of the word (even though they rely on exitus):
Suffragium de exitu Britanniae procrastinatum
Theresa May, prima ministra Britanniae decrevit, ut suffragium, quod die Martis (11.12.) de exitu Britanniae in parlamento Britanniae fieri debebat, procrastinaretur. Tum pars delegatorum factionis conservativae flagitavit, ut de fiducia illius factio suffragaretur, sed suffragio facto May votis ducentis contra centum septendecim (200-117) superior discessit.
Also, this use in Nuntii Latini confirms that exitus is a decent choice for this kind of an exit.
Consider for example: "When will Brexit happen?" "What will May do if Brexit fails?"
The word here is a noun, and I have yet to see it used as a verb in the news.
Phrases like nolite brexire are possible but come across humorous and weird.
I wouldn't say "don't brexit" in English.