I'm not sure I believe it. First, William, nor Mary, nor Anne ever even acknowledged that James Francis Edward was James II's son. Anne was the primary person who spread the fake warming pan (fake child theory) and apparently made Mary believe it (which was one of the only ways Mary salved her conscience since she did not initially favor dethroning her father). William in his international correspondence (for public consumption) with his anti-French allies continually called JFES the "so-called Prince of Wales". To adopt and admit they had stolen the throne from a rightful heir (which JFES was) is something I can't see William doing. And I can never see Anne giving up her right to the throne, which making the Old Pretender Willam's heir would basically entail.
Also I can't see Mary of Modena ever giving her son up to William, who had basically stolen her son's birthright, caused her husband to die in exile AND, against international protocol and laws, refused to return Mary's (pricey) dowry which the British govt. had confiscated.
First off, that's not actually the case. None of them acknowledged him in
public. I don't know much about Mary's opinion, but during the early 1690s a lot of people floated the idea of William and Mary adopting the Prince as their son and heir. From what I can tell Mary never really shot it down. As to William, there was no doubt in his head about the Prince's paternity. According to Lord Ailesbury, for one, William once asked to see a portrait of him and remarked "About the mouth he is most like my uncle King Charles, and his eyes are most like his mother's." That doesn't sound like someone who thinks the Prince is a fake. Hell in correspondence with Electress Sophia, who had everything to gain from the Prince being a fake, both acknowledged that there was no doubt that James Francis was the son of James II and Mary Beatrice. Only Anne never acknowledged her brother period.
Second, if William wasn't honest why make the offer? It would have done more damage to him than James II if it became public. From the various books and bio's I've read on the era and William III, its increasingly clear that William was sick and tired of England and was quickly becoming disinterested in its future. The only reason he had invaded in the first place was to secure English resources for the war with France. Lets be honest, if James II had signed onto the Grand alliance but did everything else the same chances are there would have been no Dutch invasion. William would have had his English support.
As for Anne, William hated her and would have no reason to support a woman who brought no advantages and had treated his wife, her sister so horribly. Also, before anyone brings him up, the Duke of Gloucester was basically a nonentity. Anyone with half a brain would tell that, at best, he'd be the English Carlos II and at worst die young, like he did. James Francis Edward was the best choice. He was the legitimate heir and his accession would eliminate a potent weapon from France's arsenal. Not to mention an agreement with Saint-Germain and Versailles would secure William's reign for life.
Finally, to Mary Beatrice, her opinion wouldn't really matter. However, by the time William offered to drop the Catholic part (around 1700 since he promised to veto/repeal the Act of Settlement) I think he also had dropped the demand for the Prince to be sent to his custody. Chances are the agreement would be similar to what the Tories suggested in the early 1710s: the Jacobites remain in France for William's reign and return with the new King after his death, not unlike what happened to Charles II really.
One last thing. Remember that the Tories came close to deciding to have James III succeed Queen Anne in 1714. It was only her supposedly unexpected death (the fact that she'd been sick for over a year kinda makes me call BS) that derailed those plans (the Tories being caught off guard). Here we'd have the King, supported by the Tories, angling to have the Prince succeed William. A much better situation for a restoration.
Besides wouldn't he rather adopt a Nassau relative? OTOH the only way for William to deny Anne would have been him been him having a male heir, even from a second marriage. Sure there were treaties, but if William III like the Habsburg Charles VI would have had time enough for such a hypothetical male heir from a second marriage, I'm sure he would have done anything to have him as his successor.
The House of Orange-Nassau had NO claim to the English, Scottish and Irish thrones. William's claim was via his mother, not his father, so no good there. Second, the son of the King trumps the daughter of the King. If James's rights were recognized, he'd trump Anne and any issue she had. Third, William became King over Anne because he was married to Mary. Any children from a second marriage would be behind Anne and her issue.
The fact that William never remarried even after the Duke of Gloucester should tell you all you need to know about his chances of having a child. Clearly he knew that there was no chance of his fathering an heir. Remember William had a mistress, Elizabeth Villiers, for 15 years with no child born. After the relationship ended Villiers married the Earl of Orkney and had three daughters in quick succession. Clearly any problem was with William.