Ok so, Catherine had 4 healthy children at this point, and Jasper had at least 1 if not 2 acknowledged illegitimate, so children weren't necessarily off the table. And Catherine was only 27 when they married, although he was 56. If he and Catherine had a son around 1488/89 and maybe a daughter around 1493/94, then we have an interesting scenario. These are two young people with very strong links to the royal family with absolutely no real claim to the throne. Thus, I can see these children being very highly prized in the marriage market and probable favourites of the ROYAL Tudor line.
Name wise, we have to look at how Catherine and Jasper named their own children OTL. Catherine went with fairly conventional with her choices, they were all royal names connected with the Yorks. Jasper, meanwhile, went with Helen/Ellen and Joan, both fairly simple, conventional and English names. So I can see the names being fairly conventional and callbacks to their Tudor roots. Perhaps Henry or Edmund for the boy (I personally think Edmund, if just in memory of his brother, although Owen works too) and Catherine, Jacquetta (probably not but still) or Margaret for the girl (I like Catherine better, just to be a callback to Jasper's French Royal mother.
Now where'd they marry is an interesting question. If Catherine is unattached by 1511, then the widowed Thomas Howard would make a good husband, particularly because I can see Catherine wanting some sort of Duke if she can, if just because 2 of her brothers are Dukes in this scenario. If not him, then I can see a match with Thomas Grey, 2nd Marquess of Dorset, but that's a little too close for comfort. And in 1511 she's only be around 17/18, so it could work. So A proper Tudor/Howard match. For our Edmund, I can see some form of semi-royal bride, or perhaps just a woman with some clout with the royal family. I see perhaps Margaret Courtney or Jane/Elizabeth/Margaret Stanley, or perhaps both if we have the Stanley girl doesn't survive one of her childbirths.
Now here is what I've come up with:
Jasper Tudor, Duke of Bedford (b.1429: d.1495) m. Catherine Woodville (b.1458: d.1497) (a)
1a) Edmund Tudor, Duke of Bedford (b.1487: d.1531) m. Jane Stanley (c.1490: d.1515) (a), Margaret Courtney (b.1499: d.1518) (b), Ursula Pole (b.1504: d.1570)
1a) Miscarriage (c.1513)
2a) Stillborn Son (c.1515)
3b) Henry Tudor, Duke of Bedford (b.1518)
4c) Catherine Tudor (b.1521)
5c) Miscarriage (c.1524)
6c) Owen Tudor, Earl of Pembroke (b.1527)
7c) Stillborn Daughter (c.1528)
8c) Anne Tudor (b.1530)
2a) Catherine Tudor (b.1493: d.1569) m. Thomas Howard, Duke of Norfolk (b.1473: d.1554) (a)
1a) Margaret Howard (b.1513: d.1514)
2a) Stillborn Son (c.1514)
3a) Henry Howard, Earl of Surrey (b.1516)
4a) Mary Howard (b.1518)
5a) Miscarriage (c.1519)
6a) William Howard, Viscount Howard of Bindon (b.1522)
7a) Stillborn Daughter (c.1525)