Call me an old cynic here, but does anyone else suspect that the UK Government/Parliament will somehow thrash out a deal at the last minute (well, 29th January by the looks of it) that the EU has pretty much already ruled out (e.g. Single Market access/Customs Union but with controls over free movement of people), but the EU will have little choice but to ratify it in order to avoid No Deal?
The EU are (very) obviously putting the ball in the UK court and saying that it is 'entirely up to the UK' to find a solution (i.e. to abandon its 'red lines')... (though this is disingenious, since the EU also have very clear 'red lines', and it is not entirely up to the UK at all...)
...so the UK calls the EU's bluff at the last possible opportunity, and agrees to abandoning a key red line (staying in the single market and customs union) but only if the EU abandon its own red line of demanding free movement.
If the EU don't agree, then the UK exit on WTO terms and the game ends with the ball in the EU court (i.e. the UK will have a very public vote to 'ratify' a deal, and the EU will very publicly vote it down).
No Deal/WTO terms would be bad for all involved, but in that circumstance it becomes extremely important who gets the final blame for it...
This might sound crazy, but it does at least explain Labour (and the SNP's) odd position... the outcome stated above (practically the definition of 'Soft Brexit') is what has been ruled out since Day 1 by the EU, but it may turn their own argument against them. The EU have told the UK that the deal voted down this week was 'the only deal possible' - but if the UK can agree to a Soft Brexit deal without accepting Free Movement of People (or something to that effect), then the UK can turn around to the EU and say the exact same to them.