The problem is that I'm not sure that European leaders will allow much flexibility during the negotiations or after them. Our Parliament and MPs probably has less influence over the outcome now than they ever had whilst debating EU treaties/legislation/directives/etc.
I think you are right there will be little flexibility from the EU leaders - they can either have the cake, or we can eat it, but we can't have both. So we can either retain significant access to the single market or we can restrict free movement, but we can't have both.
What the by-election does it bolster the parliamentary view that it must have a significant say in whether we have the cake, or we eat it. And also it increases pressure on the government to make clear which of cake or eating it preferred.
Now the reason why the government has been so coy is nothing to do with running commentaries, or giving away negotiating positions - that's non-sense as all that is being suggested is a headline priority which would have to be revealed at the 'talks about talks' stage anyhow. No the government is terrified at stating whether having cake or eating it is priority because as soon as it does all hell breaks loose and the uber-fragile lies holding together the brexiteers will dissolve. So if the government opts for soft brexit the UKIP hard line nutters will throw a massive wobbly, but if it goes for hard brexit a new consensus will form, almost certainly with a majority, between the 48% remainers, for whom hard brexit is the last thing they want, and the soft brexiteers, for whom remaining is probably preferable to xenophobic isolationism.