Well I've seen a few scenarios about the Spanish Armada where the Infanta is made Queen of England. She gest overthrown, but its just a thought.
Philip had a claim to the throne, however. It was a tenuous one, though arguably stronger when interpreted under Castilian law. His argument was that, as husband to a dead Queen, her claim passed to him and he ceded it to his daughter. The French had absolutely no claim to England. People in this day knew how to use the most tenuous idea to fabricate a claim, but people knew that when there was no feasible claim (i.e. the French position) that they couldn't get away with just muscling in. Besides, a Bourbon prince would only ensure pro-French control of England for a generation or two. After that, politics tended to diverge anyway. It would be inherently risky.
But your point on marriage politics. What about Charles II? He lived in France during the Commonwealth years and was a notorious womaniser, I could easily see him getting "involved" with the relations of the Bourbon Court. Something could come up from there.
It's possible, moreso IMO than the above idea. However, he would need to go a step beyond womanising to make it work. An illegitimate child would never be acceptable, except in exceptional circumstances - i.e. how his illegitimate Protestant son almost won popular support for the simple factor of not being Catholic when James II was coming to the throne. With a Bourbon princess, the child isn't likely to be Protestant. Also, the French mistress thing was interestingly hypocritical. While the French were happy to recognise and even encourage the practice of mistresses, and even gave the royal mistress an official position in court, if a princess were to have an affair it would be regarded as a matter of huge shame to her.
You'd have to have them get married, I think. And if that were to happen it might affect his credentials for returning to England. Also, the child wouldn't actually be of the Bourbon dynasty.
I guess you could play around with the Glorious Revolution. Say Charles' daughters aren't brought up so Protestant, or maybe Charles dies early and James forces them to marry Protestant, then if he dies without a son etc then Princess Mary could take the throne married to, say, a Duke of Orleans. The English would probably resist but they could be defeated. It would, of course, screw up English religion completely, but then I think any Bourbon King of England would try to go uber-Catholic.