It is too soon to tell which of the two, Brexiter or Brexiteer will become standard, or whether there will be a British-English//American English divide.
Brexiteer appeared in The Washington Post today, Tuesday, June 28th, page A19 in the column by Eugene Robinson:
Meanwhile, the other leading Brexiteer, U.K. Independence Party leader
Nigel Farage, admitted that Britain won't actually see a savings of
hundreds of millions of pounds that could be used to improve the
National Health Service.
Brexiteer was also used in the Wall Street Journal on June 27.
Ever since the U.K. voted Thursday to leave the European Union,
there’s been no end of commentary, favorable and invidious, comparing
the Leave campaign to the anti-establishment wave that swept the GOP
this spring. In one respect the comparison is apt. Both the Trumpkins
and the Brexiteers aimed their fire at a toothless opponent—and the
wrong one to boot.
Robinson and the WSJ are at opposite ends of the political spectrum.
My initial scan suggested that Brexiteer was AmE and Brexiter BrE, but not necessarily, as this headline from a Spectator blog shows:
Brexiteers need to act now, or become the most hated people in history
But the Guardian, as of June 21, preferred Brexiter
Paradoxes of a London Brexiter
As did The Times of Israel on June 27.
Lead Brexiter says no need to rush leaving EU
Brexiters and Brexiteers will fight it out over the next few days or weeks, and may the word with the most panache win. That's Brexiteer, IMO. At this point, dictionaries are worthless on the subject. I'm waiting for The Economist.