Fuel pressure issue

If you pinched the line just at the carb,that is to say pinched between the carb and all other devices and there was no change;
then the fluctuating is not related to fuel pressure nor to fuel level;
see how easy that was!
So now you have an AFR problem or a fluctuating timing problem.
To determine which, definitively, just put a timing lite on it and see what it is doing at idle. If the timing remains rock-solid, below 1000rpm or so, then you can move to the carb.

Now this is the big one, and I think this is where you are gonna find your problem; That Eddie carb has metering rods right? loosen the covers with the engine running, rotate them out of the way so you can see the pistons, but not too far and then tighten them down such that the covers still keep them from popping out of there. Check to see if the rods are properly installed and not bent, then watch them for several surge cycles to make sure they are staying down. If they are popping up and down; there's your problem. Now you just have to figure out why this is happening. See below
But if they are staying down, lets move on ;
First thing to do is to make sure the choke valve if hanging vertical and that the secondaries are fully closed but not sticking. Put a rag into the secondary side and seal it off; the idle speed might change but it shouldn't be by much. If it does; fix it!
Second thing to do is to prove all the vacuum lines are hooked up right, none have fallen off or ruptured, and this includes the PCV and brake booster hose. Just pinch them off one atta time and see what happens. Don't forget to plug any un-used vacuum ports on the carb.
Next you check that the carb base gasket is correct and sealing.
And finally check that the intake-ports are not sucking air at the heads.

As to the metering rod pistons; if they are jumping up and down, this is in response to the vacuum falling as the engine is slowing down and preparing to stall.And then rising as the metering rods allow more fuel to flow.. This is usually caused by a plugged or poorly working low-speed circuit, or severely retarded timing. My go-to is to just grab the Vcan and pull some timing into it, and see if it likes it. If yes then pull some more into it. Then slow the engine down some. Now you have to figure out the why of that.
But if the surging continues in any way, even at the higher rpm;then I would suspect the lowspeed circuits.
Don't forget to pull the rag out of the secondary side,lol.
Good luck.

Oh wait; I'm assuming the exhaust system is idle-flowing adequately,lol