Cross post.
fid, it'd be absolutely plausible, because that is the sum total of what you need to do to contract a marriage in that period.
You just need to swear that you have done so. You do not technically even need witnesses - that's just for convenience in case you're challenged. Nor do you need to be in church.
You get all sorts of fun legal cases where people quibble over precisely how formal their swearing was, claiming they didn't mean it seriously and didn't use a proper, formal declaration.
There is a brilliant arse-kicking woman in the century before this one, who privately contracted a marriage with the family bailiff. Her mum dragged her in front of the bishop of Norwich, and got him to give her a lecture about duty and obedience. And he says (in effect) 'look, be a good girl, we'll get you out of this, just tell me the wording you used and I'll tell you it wasn't solemn enough, and it'll be fine'. And she replies, 'thanks, now you've told me how solemn it needs to be, I'll go do that, so I'll have confirmation from a bishop it was done proper'. 
She gets to stay married and they have three (?) kids.