Dale. I think conflated is the right word. Confused would be another. So far, it looks to me like these are different families. Genealogists have grabbed one, then grabbed another, then come up with idea that they must be the same family even though there's no evidence. Not every William Rice is related to every other William Rice.
If you think there's a missing Tudor prince to be found, then you've already misjudged the way to research a question like this. A missing prince might emerge from the evidence, but not from smashing all the families together into a single family and insisting that he's one of them.
If you're looking for a Tudor prince? What's wrong with John Perrot? He fits the bill. He's probably related to the family of Perrott Rice. The only problem is that his mother was a baroness, not a laundress.
Another thing to remember when you look at something is that these people were not just modern people dressed in funny clothes. Their society was very different from ours, and they were extraordinarily more class conscious. You see great nobles with grand titles like Chief Butler and Earl Marshall, and imagine that ever butler was some kind of noble. Not so.
The laundresses at Greenwich Palace were peasant women, who had to work for a pittance. To the extent that they had a chief, he or she would have been under the supervision of a many times sub steward. There was no great and honorable post as Great Supervisor of the King's Laundry, occupied by a genteel woman. No, not at all. Even if there had been, it would have been a man's post, not a woman's. And, that man would have been so far down the totem pole that he never would have been able to marry into the gentry.
When Henry VIII's father wanted to humiliate the pretender Perkin Warbeck, he put him to work in the scullery. Even if the poor had been royal, that effectively put an end to any aspirations. Not just because he was defeated and captured, but because it placed him so far down the social scale the he could never get back up.
You're looking for the laundress' family among the gentry. That is never going to work. As the Paston Letters make clear, a manorial family would never tolerate a daughter marrying out of her class, not even to a prosperous steward. The Paston daughter who defied them was cut off from her family and her inheritance. She was, quite simply, dead to them.
It's not beyond reason that Henry VIII had a night of passion with a laundress, although more likely 10 minutes. Maybe the laundress went to her grave swearing it was the king's son, but given the sluttish reputation of laundresses, it's unlikely anyone but her mother would have believed her. And, you can bet that if she noised that around the palace, she would have been whipped and fired for spreading such lies. No manors or royal preferments for her son, no secret deals, or doting royal half sisters.