Sorry @Gregory and @Cristiano, the flag is likely Spanish! I should have mentioned, the sitter's name is inscribed in the corner of the portrait and is definitely Spanish, not Austrian. The sitter is gesturing to this ship and is sitting at a table with a sextant, compass, and (seemingly) generic map. I searched each of his family names in various online genealogical databases with no results, so I think I will need more detail as to his geographic origin before narrowing down any physical archives that may contain record of his life...or his ship.
I assumed the date of early 1800s (likely 1810-20) because of the sitter's clothing- men's fashion at the time was becoming homogenized on both sides of the Atlantic and the trend cycle was moving faster than in previous decades, allowing for a more narrow window of dates. But if the ship really does look like something from the 1830s-40s @wefalck...I suppose he could either be terribly out of fashion or certain details were painted or restored inaccurately.
I appreciate you all for your thoughts and assistance!! Even knowing that this type of ship is rather common in this period helps my search, so thank you!