My fancy bike pump is gummed up with the tire goo from tubeless tires - help?

I got a new bike last fall and it's my first time with tubeless tires. They come with the goo inside the tires that "heals" any nicks or cuts, to help with not getting flats and quickly repairing flats.

When I unscrew the valve to inflate the tires, it sputters a little and some of the goo oozes out. I didn't pay much attention, but now I can hardly get air through my fancy Lezyne metal floor pump, even when it's not attached to the valve at all. Just lying on the floor, I can hardly pump air through it. So I have to assume some of that goo made its way into the valve end of the bike pump and dried in there.

What to do?
On the Lezyne, I can actually unscrew the end. It's one of those ends that is presta one one side and schrader on the other side. So I could get it off and soak it in something, but I fear that whatever would get the goo out, would also disintegrate the rubber or silicone or whatever makes it work.