between Hilsea and Portsmouth” might imply the Green Post or the Fountain, or a pub even further south at Portsea.

Quote

A general Court Martial assembled at Portsmouth, on Friday, for the trial of several soldiers of the Inverness militia, for rioting at a public-house between Hilsea and Portsmouth, and striking an officer of the 42d regiment, in the execution of his duty; Lieutenant-Colonel Carne, East Essex, President.

Star (London), Tuesday 05 January 1813 source

Quote

The General Court-Martial, of which Lieut.-Col. CAINE was President, for the trial of several soldiers of the Inverness Militia, at Hilsea, have closed their proceedings.

Pilot (London), Tuesday 19 January 1813 source

The name of the pub may be in records of the court martial, if they still exist. I’m sure I’ve read a more detailed account of a court martial related to a Hilsea pub - something about treasonous speech - but I can’t put my hand on it right now.

Some possible leads: