I didn't see any of the weapons you mention
I did see the 1903 Springlield
The Lewis Gun which is air cooled, was an issue weapon, as was the BAR
The P-17 was in the US arsenal, but after WW1 was rarely issued to units, held in reserve so to speak because it took the same caliber and loading system that the 1903 Springfield, 30-06 and stripper clips. After WW! The P=17 was relegated to storage , in favor of the o3 & 03A1 I consider the P-17 a better rifle
The Thompson while in service probably NOT issued to a backwater like China
I am surprised that shotguns were not in evidence, the Winchester 97 or model 12
You don't have to stand tall, but you do have to stand up!

The Navy did have Tommy guns at that time, but none were used in this movie. In the book the sailors also use shotguns to repel boarders, but they do not in the film version.
"Pray for an early spring or permission to open fire."

This movie was highly accurate, from what I know is the Marines bought some tommy guns in the 1920s. Maybe the Navy never had them until WWII.

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In the book there is reference to "riot guns" being issued for repelling borders and for at least one shore attack against pirates. Probably Winchester Model '97s, one of may favorites.
I do not know why the USN chose not to arm these boats with belt fed machine guns as you described but they did not.

USN loved the Lewis. Army hated it. Higgins Boats were frequently armed with the Lewis right up to Guadalcanal.

I've read that the Lewis MG was common in the US Navy right up to the early part of WWII.

The Lewis guns are absolutely correct. Kemp Tolley mentions Lewis guns several times in his book "Yangtze Patrol". Newer editions of the book even have several pictures of them, some showing them in the armored mounts on the gunboats, and one of a landing party with 2 guns on bipods. Tolley also mentions that shotguns and Thompsons were issued to gunboat crews.

Im surprised they did not have shotguns in the small arms lockers.
Is the use of the Lewis gun instead of the M1919 or the M1917 machine guns correct?

M1917 Enfield rifle in US military service from 1917-1953. Thompson submachine gun in US military service from 1938-1971. However the Thompson was available starting in 1919 and as you could see the crew of the San Pablo kinda did whatever they wanted to. So it was possible, though not probable, they used their own personal ones. The water cooled Browning M1917 Machine Gun and it's varients were used from World War I to the 1960's until replaced by the M60 using the 7.62 mm standard NATO round.