Lawmakers will this week hold a nonbinding vote on the future of Britain's trade after Brexit. It is a vote that could set in motion a series of events that could lead to the fall of the Prime Minister and her Conservative government.The House of Commons will vote on whether the UK should stay in the customs union with the European Union — allowing tariff-free trade to continue with the bloc after Brexit.While the vote won't be legally binding, it carries huge symbolic weight, because it will show the strength of opposition among members of Parliament to May's plan for a hard Brexit — which would see the UK leave both the customs union and the EU's single market.Leaving the customs union is one of the Prime Minister's red lines in negotiations with Brussels. If the vote goes in favor of staying in, it is a serious test for May's premiership.Last week, the House of Lords voted against May's government and in favor of Britain staying in the customs union — showing that there is no majority for a hard Brexit in that upper chamber.Potentially, there is also no majority for a hard Brexit in the House of Commons. What's more, the Prime Minister has no overall parliamentary majority for her ruling Conservative Party, which is only in power because of the support from the smaller Northern Irish Democratic Unionist Party.So May would have a dilemma: does she continue to maintain her red lines on Brexit, or does she start to recognize her political weakness and make concessions?This is where it gets dangerous for the PM. Some of the most powerful members of her Cabinet are leading Brexiteers — like Foreign Secretary Boris Johnson, Environment Secretary Michael Gove and Brexit Secretary David Davis.The pro-Brexit group of backbench MPs, led by Conservative MP Jacob Rees-Mogg, also holds sway in her party. Any move by May to water down her position on the customs union could provoke a leadership challenge by pro-Brexit elements in her party.The matter is therefore turning into a game of chicken for the PM, and will come to a head when the Commons has its first opportunity for a meaningful vote on the customs union — likely next month.Some British media has suggested that this vote may become a confidence vote: meaning that if the government loses the vote, it has 14 days to win a confidence vote. If it fails to do so, Britain will hold its third general election in four years.On the face of it, this seems to be a warning to Conservative anti-Brexit rebels: back May and vote against a customs union, or else risk bringing down the government and let in the opposition Labour Party.This high-risk challenge — effectively a call by May to "back me or sack me" — could pay off for the PM. Conservative MPs do not want to risk another election.But as the Prime Minister knows from bitter experience after calling a snap election last year — which resulted in her losing her majority — high-stakes gambles can backfire spectacularly.There is also a cold, hard calculation that Conservative MPs can make. For those who wanted Britain to remain in the EU, securing membership of the customs union would mark a significant weakening of Brexit and, in their eyes, lessen the potential damage to the economy.They could rationalize a vote against May as acting in the best interests of the UK. There would also be a buffer or cooling-off period between the PM losing the vote and another general election being triggered, under which the government would have two weeks to muster support in the Commons. Under those circumstances, the current government could be saved — but it is almost certain that May would have to stand down.The Prime Minister does not have huge personal support among her MPs, and there are many on both the pro- and anti-Brexit sides who would like to see her gone from Downing Street. She should be wary, therefore, of making the customs union vote a matter of confidence.There are also signs of compromise elsewhere: on Wednesday, her Brexit Secretary told the Commons Brexit committee that a final motion among MPs this autumn to approve the entire Brexit deal reached with the EU could be amended.This is a change from the "take it or leave it" offer from May.And Bloomberg reported this week that the UK could have a say on trade if it stayed in the customs union — the sort of bespoke arrangement that May has previously suggested might work.The Prime Minister does not have to take Brexit — and her own political survival — to the edge of the abyss. But to regain her authority over Parliament, she might just do that. Right now, I wouldn't like to put money on where any of this ends.