Well, he could always pull the same thing he did with Elizabeth Woodville with one of his other mistresses.
After all, Richard III's whole claim to the throne was from a claim that Edward had promised to marry Eleanor Butler. She's from a noble family (better than the Woodvilles) so could maybe pull it off if he did promise to marry her and she could prove it.
The claim was that Edward
had married Eleanor, not just
promised to marry her. And in those days promising to marry somebody, followed by physically consumating the relationship, meant that legally -- even without having had a religious ceremony -- you
were married, anyway. Admittedly the Church didn't like it, but such 'common law' marriages remained fully legal in England until the 18th century.
Edward IV's OTL marriage was received with something far less than thunderous applause. Had there been any basis to question it, why was it not brought up until after he and Eleanor were dead?
Allegedly the only witness who could have revealed the truth, apart from the couple themeselves, was the priest who'd solemnised the relationship for them and he didn't pluck up the courage to come forward earlier. Some Ricardians think that he told Bishop Stillington and possibly Clarence about it during the later years of Edward's reign, if Stillington himself wasn't that priest (which would have been feasible), and that Edward hearing of this -- probably from Clarence, privately trying to use it as leverage -- was why Stillington was confined as a prisoner in the Tower for a while around the time of Clarence's own imprisonment & death. As far as I know there's no
other reason given for Stillington getting locked up.
Butler is the daughter of an English earl and respected military commander, so I could see her as having the standing to insist on marriage if she tried. She doesn't seem to have done so IOTL
The daughter of an earl whose son & successor, her own brother, had recently died fighting
against Edward... so less leverage than would otherwise have been the case. From the fact that Eleanor retreated into a nunnery not long afterwards, it's suspected that any relationship betwen her and Edward was probably one in which she simply hoped to win Edward's favour for her family (rather than see therm penalised as his enemies) rather than to become his queen, or even that she felt coerced by circumstances and was happy to leave once he'd tired of her. That still doesn't rule out Edward having made the promise as a way of gaining her favours -- perhaps in the expectation that if she went public then he'd be able to convince people she was lying -- though...