The USCG enjoys federal police powers ashore and off shore US Military powers. Just about as much power as any individual can get. They don't require a warrant to search and they usually carry miltary weapons with firepower on board their vessels like most small US Navy
vessles. They don't carry nukes.
About the most you could actually do is file a complaint after they are gone. That may seem like a little but the USCG enjoys a pretty decent reputation. They generally are more polite than a police officer would be and they do follow up about complaints were you to file one. The don't usually get involved with random searches so they usually have a mission at the time they stop you. They often are looking for someone or some vessle they will ask questions and conduct searches. If they thought you were involved in something or knew somwething they could detain you or part of your crew.
If you give them a hard time they will return sufficient force as required. using a wepon would be a bad idea. They don't need a warrant to search your vessle and they won't accept a refusal when they request you to do something. Your rights are limited if your cooperation is minimal.
FWIW, if in the area they will rush to save you should you require it. It does cut both ways.