Your POD is too early for a start. It took James II (the OTL one) several years to properly anger the British public enough for them to openly advocate Monmouth's position.
As for whether it could happen...undecided. I'm leaning towards "no" as generally rebellions aren't the sort of thing that could cause civil wars - they tend to be either dead certs for success, or woefully undersupported to meet their ends, resulting in either rapid success or rapid failure. To be able to cause a full civil war, you'd need Monmouth's cause to be popular enough that his supporters would be willing to not only turn on their King but drop everything for several months or years and focus on winning a war - this means creating not a rebel force but a proper army, with not just a stash of weapons but the facility and finance to make more. You also need the King to be popular enough to do the same. You also need the issue they are fighting over to be ideological (which it was) and fully black-and-white) which it wasn't, given James II's repeated claims that he was respecting the Church of England) because after a few months, the population is going to get sick of the war and be willing to compromise their beliefs in order to secure peace. Only if the war is fought over an issue where it is impossible to argue a grey area in the middle which lets you, say, support the King in return for a return to peace and your normal way of life, can a civil war continue.