There's the rub. England can invade but they're one lost battle/dead king away from political turmoil. Can anyone versed in 15th century English history suggest who would become king of England if the Lancasters are tossed out?
The Mortimers. Edmund Mortimer, 5th Earl of March (1391-1425) was heir presumptive to Richard II, by descent from Edward III's second son Lionel. His uncle (also Edmund) claimed the crown for him in the rebellion of 1405 (uncle Edmund having married Glyndwyr's daughter). However, the younger Edmund remained in royal custody, was raised in company with the future Henry V, and was Henry's loyal vassal. When the Southampton plotters approached him about displacing Henry in 1415, he immediately told Henry all about it.
Edmund had no children. The Mortimer claim passed to the House of York through Edmund's sister Anne, who married the Duke's younger brother. (He was one of the Southampton plotters, and Edmund sat on the court that condemned him to death.) The Duke died at Agincourt (without issue); the dukedom and the claim both passed to Anne's son Richard, who launched the Wars of the Roses many years later.
There were several Lancasters around in 1400-1410: Henry IV, of course, and his four sons: Henry (1386), Thomas (1387), John (1389), and Humphrey (1390); also two daughters: Blanche (1392) and Philippa (1394). The latter two married out of the country in 1402 and 1406.
If the Lancasters are
collectively "tossed out" - that's going to be weird, because there are so many. The next in line is Edmund, who I think has to be the replacement, but before 1409, he's a minor. His political survival up to that point has been achieved by his apparent loyalty to Lancaster. I don't see how he could suddenly disclaim that loyalty and become a rival claimant without getting locked up and probably abridged.