Got it! Once you let me know that it had to go in without the plate coming out, I was able to make it work. Basically, I'd been feeding the outer edge of the gasket in first, which was causing the gasket to stay firmly in a "sideways" position around the plate. I realized I had to feed the inner edge of the ring first. That left the gasket in a less stable sideways position, (like it "wanted" to flatten out) allowing the portafilter to flatten it out and force it into its proper place. I have no idea if this description makes any sense at all, but the upshot is... it worked.
Thank you very much for your assistance. I didn't want to mess around too much in case I was missing some obvious method to remove the holding plate. But knowing that it had to work without removing the plate was a HUGE help. The bike tire analogy was very helpful as well.
I can now stow my stovetop espresso maker away until my next camping trip.