There's a lot of variables, deck vs keel stepped, backstay or no backstay, single chainplate vs multiple chain plates.
When not under load from sails mast can be remarkably stable with minimal tension on the shrouds and stays. If the shrouds are attached to their own chain plate, i.e., 3 chainplates per side one each for forward, aft, and cap shrouds, then as
@Stu Jackson notes one shroud at a time can be disconnected without using a halyard to stabilize the mast.
On the other hand, if all the shrouds attach to a single chainplate, as is the case on my boat. It would be necessary to use a halyard to stabilize the mast. Although other considerations may be dictate taking the mast down would be a better course of action. My chainplates are secured with 14 bolts each and 4 feet long, with a couple of bends. Because of this I opted to remove the chainplates with the mast down. Even then it took the better part of a day to remove, inspect, clean and reinstall each chainplate. Removal was particularly challenging because of the copious amount of caulk involved and the shape of the chain plate.
The short answer is it is possible to remove the chainplate with the mast up, but each boat model and boat is different. And if some fool used 5200 on your chainplates, it will be a long struggle. For those of us not familiar with the H45CC photos of the chainplates would be helpful in providing more specific advice.
Good Luck.