If you let it run on for a bit longer, the idea of a second referendum would become more palatable to both MPs and the public. And honestly, a second referendum would be the only thing that can get us out of this mess without devolving into a complete shitstorm and/or constitutional crisis. Allow an STV, place down the options of no deal, an already worked upon and proposed deal (sticking it on a referendum's more likely to get the MPs to agree to it, failing that, a generic "leave with a deal" option), and remain (And if anyone here is about to complain about how that gives remain an unfair advantage because it's split the leave vote, that's why I'd recommend an STV. It stops vote splitting being an issue). That way things can go three ways:
1) We get a majority for no deal. We crash out. The economy and union are both fucked, but at least we avoid the political nightmare. Some people are upset, but at least the majority are content (unless things go really sideways for the economy).
2) We get a majority for a deal. If it's a specific deal, great, it's answered. If it's not, then the remain and no deal MPs are more likely to fall in line with the idea of agreeing to at least something. Again, some people are upset but this is going to probably have the fewest upset people. No-dealers are more likely to see this as at least *a* brexit, and remainers are going to be happy it's at least not a no-deal
3) We get a majority for remain. We remain, some people are upset, there are cries of "Oh so you were just going to do a referendum until you got the result you want?" but ultimately nothing comes of it.
In essence, a second referendum will give us a true mandate. 52% was a very shaky mandate in the first place, turning it from "We're definitely getting a deal!" into "Look you all voted for no deal" basically completely delegitimised the mandate because the referendum never specified that, and 90% of the leave campaigning was saying that we'd get a deal.
Obviously it won't work out this perfectly, but really all the other alternatives involve utterly alienating half of the electorate and potentially slamming one of the first nails into parliamentary sovereignty's coffin.