I'm not really sure what you mean when you ask if Charles would be that stubborn. It's not really like he had a choice. England wasn't an appointive monarchy.
James was next in line for the throne after the childless Charles. OTL, he ascended the throne with a massive amount of political goodwill and proceded to throw it away over the next few years.
Mary, with William attached, is the next protestant heir so when/if James does his runner, they get the throne as OTL. If Mary, William of Orange and William, Duke of [not Gloucester] all die on time, leaving the childless Anne as Queen and Henry of Gloucester as the heir apparent and King in 1714 or so.
Of course, with sixty years between the POD and the accession, there is room for a lot of butterflies in this scenario. Personally, I like the idea of an uber-loyal Henry facing down William of Orange and rallying support for his nephew, James III/VIII in 1688...with him as regent and no Glorious Revolution.