They've already said that they will, as an alternative to a permanent presence. See Gove's letter to the leaders of the Stormont Assembly. Paragraph 3.
And I don't think you need to open our eyes about what a bunch of untrustworthy shits they are. We've lived through this. But as for "the court of public opinion", their MO is far more to meekly cave in and just shout loudly that they haven't.
They've rightly assumed that by the time it next matters, their potential voters either (a) didn't care anyway, (b) have forgotten or, the biggest group by miles (c) have never heard of any of this because they don't really care one way or the other.
Can't imagine they'd need to get CJEU involved. As you rightly say, the terms of the withdrawal have been written into UK law. Just pursue the matter through the British courts.