pull that oil filter, cut it apart and have a look see of whats in it and also look behind it (that adapter with the check valve that may be whistling as oil is bypassing). if you find filings in the filter element then you KNOW it's internal and need to stop running the engine since that will cause more damage. that is the quickest and easiest thing to check and causes no more damage if there is an internal issue.
do you have any specs on whats inside the engine? anything different from a stock 350, windage trays, high volume oil pump (which may require a high flow filter) or special timing gear set up etc? have you tried simply swapping the oil filter or listening directly AT the oil filter? some filters have an anti drain back valve built in and this could also be "whistling" as oil goes through it,theoretically. it would be worth a listen maybe. are you sure the oil filter is the correct one for the engine? some filters have the same dimensional size and thread but are made to flow opposite of other filters, so from the inside to the outside or opposite to what a standard 350 would flow. if you have the wrong filter and the filter has an anti drain back valve, then the valve could be fluttering causing the noise or the filter could be bypassing due to low flow through the filter because the filter isn't made to flow the wrong way. you say there was lots of oil flow when you primed the engine but if the system is bypassing it may still have good flow but the oil will not be filtered properly. the bypass is built into the oil filter adapter and is made to open if there is low flow through the filter from cold, thick oil or the filter were to plug up (or you simply have the wrong filter). if a high flow oil pump were used or the builder changed the pressure relief valve setting in the oil pump, then a high flow oil filter may be required. these typically are for race engines where the owners sacrifice filtering for flow.
is there a reason why the engine won't stay running/idling by itself and needs to be babysat? possible cause of the noise as well?
from memory, does the noise increase exactly with engine rpm or stay the same pretty well or only a slight difference with an rpm change? if something were rubbing on a rotating part then it would change exactly as the rpm changes. if the noise is something like, say, a vacuum leak then it would change with rpm but not necessarily be exactly as the rpm changes, if you get my drift?
here is a little video on the oil filter system
https://www.onallcylinders.com/2016/...filter-bypass/
here is a video or 2 showing the filter cut away so you can see the anti drain back valves and what the filters look like inside. some filters have cardboard end caps instead of steel so they would be the lesser quality ones, in my mind.
https://www.youtube.com/watch?v=HXGHA8DUhGQ
https://www.youtube.com/watch?v=X3JulJDNpeU
https://www.youtube.com/watch?v=lrAbxCZlZpE