As far as next week’s votes go, how the Labour leadership feels about the European Economic Area is entirely unimportant. The big problem that Remainers have is that there are too many Labour MPs who want to vote against keeping the United Kingdom in the EEA and Conservative MPs value “not voting on a motion brought forward by the opposition frontbench” at a higher premium than “keeping the United Kingdom in the single market”.
Opposition to the EEA outside the frontbench has two causes: migration and sovereignty. A good rule of thumb is that if you were an opponent of the European project before the referendum and you oppose the EEA, it’s because you feel that it requires too great a loss of sovereignty to be acceptable, both as a fulfilment of the referendum result and from a policy perspective.
If you were a Remain voter before the referendum and you oppose the EEA, your objection tends to be that the Brexit vote precludes any deal that includes the continuation of the free movement of people. (Stephen Kinnock, one of the advocates of the EEA approach, has been arguing to various wavering MPs that this is not the case, which per my reading of the European treaties is not true, but seems to be winning over some people.)
Team Corbyn aren’t exactly lying awake at night worrying about the free movement of people, but they do, in the main, believe that the referendum result means that it cannot continue, and that the EEA does not facilitate it. The opposition to becoming rule takers is more deep-seated, however: the leader’s office fears it would limit their long-term policy ambitions. Also a problem: the nations of the EEA have a different customs arrangement to the nations of the EU, so joining the EEA would not in of itself resolve the problem of the Irish border.
Of course, to reiterate, this is all secondary to the bigger problem, which is that you need, at absolute minimum, 20 Tory rebels to pass the EEA amendment. And it is not clear that even that very optimistic bar will be cleared next week.