Or they could full: "I don't care if he's a bastard, blood of John of Gaunt still flows through his veins" and crown Charles Somerset, he's an adult and proven battle commander.
I wouldn't say he's a proven commander tbh. There's barely anything about how he got his knighthood and he kept his head very down during the Tudor reign.
I think we can all agree that the OP leaves a very very very messy situation to be resolved.
The mostly Lancaster party have won but don't have a legitimate adult male of Lancaster descent and nobility.
The definitely Yorkists have lost but are divided among several claimants that could rule or be married to a ruler.
Lancastrians:
Jasper has the potential power but not the Lancaster lineage. Free to marry the Stafford widow.
Edward Stafford has the lineage and nobility but is a minor, as is his younger brother.
Thomas St Lawrence has the lineage but is likely a minor and isn't nobility
Margaret Beaufort, mother of the winning claimant Henry Tudor. She'd have a shot at being the Lancaster Queen Regnant if allowed. However unlikely to have more children.
Yorkists:
Edward of Warwick, young but with title under attainder, cousin of EoY
Elizabeth of York, eldest daughter of Edward IV, has a shot at being the York Queen Regnant if allowed.
Lincoln, cousin of EoY, male noble of lineage but least senior.
Other:
Thomas Stanley, King of Mann, is married to the mother of Henry Tudor, Margaret Beaufort, but they're unlikely to have children.
All the combinations have flaws. There will probably be need to have a back up line that will needed to marry into the next heir.