I cannot imagine any reasonable designer would design a system where the drain plug pull would pass through the water flow piping, even the non-pressurized piping, i.e., the post valves mixing chamber. In my bathroom faucet with a lever-operated pop-up drain the actuating rod is behind the spigot.
But it appears that for this faucet the pull is coaxial with the riser of the mixing chamber! The water must flow in an annular space around a central open space that the pull passes through. Somehow water is getting into this central open space.
You would have to look at the diagram for this faucet to determine whether there are replaceable seals (such as O-rings, bushings, or other elastomer elements) which seal the mixing chamber on the bottom or the top. It may be a failure in a non-repairable press fit.
I would first try snugging up the nut on the bottom which clamps the plastic fitting onto the faucet. But do not overdo it. If this does not work you may have to disassemble the mixing chamber.
In that case, I would remove the actuation rod, loosen and remove the nut, and pull off the white plastic fitting and see if there is a replaceable seal on the bottom. There may be another on the top.
My first thought is that the actuation rod itself would NOT be sealed with O-rings, but there may be some short-cut designs currently sold which do things this way.