Just to clarify... With the seals installed, It will be fairly snug in the cylinder bore. Below is my Method that I have used for years and passed onto many members.
1. Install the Vees, very gently tighten the nut to hand tight.
2. Lube, I have a cup full of oil and I simply immerse the entire end of the pump piston.
3. PULL the pump piston Into the cylinder, it will take a gentle tap on solid surface to fully clear the cylinder.
4. Install cross pin and believe it or Not if you get off spline it can cause enough expansion where it will bind the pump piston in the bore at the upper end. This is why I recommend marking the piece and installing it in the same direction as it was removed.
5. Push the pump piston to the upper limit, install the spring, caging and C-Clip.
6. Mount the pump piston assembly into a vice on the hex portion, where you can push down on the piston and adjust the nut to where there is good resistance but not enough to where the spring can't fully retract. It's a delicate setting that can create enough vacuum to pull oil off load and not ruin the seals under pressure when they expand.
Does the pump piston pressure increase when the pin is installed? I wonder if #4 is your problem.
I put the Vees in finger tight. Lubed then up good and pulled the piston in from the end. It takes more than a gentle tap to clear the end. Once the piston is in, I can't move the piston either direction by hand. The only way to budge it either direction is to rap on each end with a dowel, or if I run a punch through the hole I can just barely get enough leverage to move it by hand without rapping on the ends.
I then tried the piston without seals thinking maybe I deformed the piston or housing.

The piston starts sliding in freely.

When the end of the piston reaches the end of the slot I get a lot of resistance and have to give the piston a good rap on the end to get past this point.

But once the piston gets into the working range it moves smoothly with no resistance.
So without seals, the piston moves freely through the working range with no binding, but with the leather Vees it binds. I wonder if maybe the housing is a little deformed, not enough to affect it without seals but with the extra resistance of the seals it binds?
As an additional test/thought I tried using one of the other seal types that came with the kit. There was an O-ring backed U cup (I think that's the term for it) that fit in the pump so I tried that. The pump moved smoothly and easily with that installed in place of the Vees so I tried putting it in the jack. It was able to lift the arm of the jack with no load, but when I tried it with a load, oil shot out of the pump. I assume that's because the U cup had no way of sealing around the threaded stem.
So at this point I'm stuck. Should I try to get a different set of Vees and see if that's it? Is there a way to use that U cup and seal around the stem? Or should I just give up on this pump and try to replace it?