You didn't mention this, so one other possibility...a long shot.
Many RVs have a pickup point for RV antifreeze just prior to the pump. The valve functions much like the bypass valves for the hot water heater.
Turn it one way and it draws from the fresh tank. Turn it the other way and it draws from a jug of RV antifreeze.
My new RV has this, and it's a first for me...never had this in previous rigs. It took me a moment to figure out what was happening.
The behavior you mention - short burst of water followed by a trickle then nothing - sounds familiar.
I was sanitizing my new rig for its maiden voyage. I have an "in-line" chlorine dispenser for my main feed hose from the house. It attaches to the hose bib and chlorinates EVERYTHING from the house to the grey-water dump. So, the whole rig was full of water, and all air was purged. Things were going great, or so I thought.
After sanitizing through the city water line, I filled the fresh tank with chlorinated water, and let it "steep" overnight. The next day, I began working on it. But the pump "didn't work." I turned on the pump with all valves closed, and residual water from the city water that had fed the lines was still in the PEX. The pump literally built air pressure and was using that air pressure to push residual water out of the faucets each time I opened a valve...but only an intermittent sputter.
I opened the lid in the dinette (where my water pump is) and saw the valve and pickup hose...again, a first for me. I scratched my head for a moment, then I turned the valve 90 degrees and bam, water!
The pump was sucking air - pumping air - through the antifreeze pickup hose and pressurizing the PEX just enough to give me a good burst of water followed by a trickle. Close the valve, reopen and another burst. Eventually much of the water left in the PEX had been blown out by the air from my water pump.
Again, it's a long shot. But if your pump is relatively new and you're kind of new at this, you might double check this valve.
ALSO: Assuming this doesn't solve the problem...while you're messing with the filter screen, see if you can help prime the pump by pouring a small amount of clean water into the filter housing before you reassemble it. Turn on the pump and use a turkey baster, catsup squirt bottle or similar tool that can dribble water slowly into the housing. As the pump picks up the water and pumps it away, having a decent amount in the tool will enable you to keep feeding more water. If the pump is bone dry, feeding it a bit of water may freshen up the seals in the pump and get it going. Once primed, the seals will get better and better with use. I had to do this once to resurrect an older pump.
Note that often times the filter/screen housing is not well oriented for this trick, but you should be able to twist the housing one way or the other so the open housing is pointed "up" and will accept the water to prime the pump.
To be fair, that old pump was in my previous rig - the PUP - and it eventually gave up the ghost. It pumped water, but the check valve couldn't do the job anymore, and it cycled on and off every few minutes as the pressure leaked back though the check valve. But that didn't prevent the pump from priming and pumping.
NOT RV Related:
Years ago I had an 1837 farmhouse with a well. The well had a "jet pump" (the pump was not in the well, but much like the RV pump, it sucked water from the well rather than pushing it up from the well as in-the-well pumps do). It, too, had a check valve -- at the "foot" (bottom) of the well. Between the pump being old and the seals worn, and the likely possibility that the check valve was leaking a bit, and the fact that the well could run "dry" now and then...the pump could suck air...and lose its prime. I had to keep a 5 gallon container of water at the ready to prime the pump if it couldn't catch prime. Fun times.
But I digress.