Chainsaw Repair

I am working on the oiler on my 1982 vintage McCulloch 10-10 saw and have torn it down to expose the oiler piston pump.  I removed the little piston pump (2 screws) and think the problem is the combination oil pickup / wafer check valve seat has hardened and been scratched over the years.  I found a new one, but the problem I am having is how to shove the seat end through the intended hole bored in the oil tank.  The diameter of the seat end of the pickup hose is a good bit larger than the hole it has to go through, and I'm thinking I need to find a way to compress the thing so I can shove it through with brute force and awkwardness, without damaging the seat face.  So far, no luck at all. See pics.

It would be easier to buy a new saw, but this one still runs good and nobody makes a saw built this well anymore.

I'm wondering if these molded rubber parts have just all become stiffer with age.  No telling how old the replacement is that I found on ebay - probably been sitting in a repair shops parts inventory for 20 years or more.  I'm wondering how they installed these at the factory.  I'm thinking like a tiny version of a piston ring compressor, but maybe it was just more pliable and could be shoved through by hand.  If I can;t get through any other way, I'm considering cutting the pickup tube suction screen off, then splicing back on afterward with a piece of larger tubing used as a sleeve.  I just don't like kludgy solutions!

Thanks for the input. I also noticed there were at least 2 versions of this, since my father had an even older 10-10 from the 1960s, no longer running that I can salvage some parts from.  His was the thinner version with phenolic disc - mine was the thicker version with aluminum disc (Pro MAC 10-10S).  I almost got a thin version through using a technique like you suggested, but not the thick one.  So I finally swallowed my pride and cut the suction hose, put it through from above, then reattached the two pieces with a sleeve of tygon tubing.  I also salvaged a pump push rod from the old saw to put in mine, as the pusher knob had broken off on mine due to "stressful" conditions while using the saw.
Since accessing the piston pump requires a major disassembly, I was a bit concerned that everything might not work after getting it all back together, but I now have a working oiler and it runs as good as before.  I had already been evaluating new saw models for replacement, but nothing I saw on the market was built as well as my old McCulloch, so I'm really pleased that I can keep cutting with it.