No, Ive run my
boat, and a number of others held by LLCs, or other forms of corp entities, across international borders many times with zero issues. The most Ive ever been asked for is a letter of authorization from the corp entity...and not often for that. I keep copies of the authorization letter, signed by me, authorizing me to operate the vessel in the ships papers.
Re
insurance, the owner (the LLC etc) is normally the named insured and additional insured parties may be added (such as the individual/captain who actually operates the boat).