Genuine question: would it be such a bad idea for the government to form a bipartisan Brexit negotiating team? Get in some of the elder statesmen and women on a cross-party basis (Brown, Heseltine, Cable, Salmond etc) and try and do it properly and moderately. Then have a referendum on the terms after the negotiations are complete?
The negotiating team could get on with it on a cross-party basis, regardless of whether we have another election in a few months, or what party is in power. It's such a big deal for the future of the country, it needs to be elevated above the level of party politics, especially as no party can command the confidence of the country.