Assembly complete! The first few test impressions were noticeably cleaner than with my previous methods, but made it very obvious that I was going to need something to help with registration. I looked back through the different ways that various presses have solved this issue and what I needed was a tympan.
A tympan is usually made from oiled paper or sometimes cloth stretched over a frame. Gauge pins are pierced through the tympan to hold the stock in place for printing, and additional packing can be placed between the tympan and the platen to produce an even impression.
I already had a make-shift chase built from a frame of 1x2" boards mounted on 1" plywood (since I lack proper lockup furniture, having a base was essential). A strip of 1/4" board ended up being exactly type-high when stacked on my chase. This meant that I would have to route channels for L-brackets to hold the tympan frame together. Unfortunately, the best I could find as far as screws to hold the frame together were M4-6 machine screws that didn't sit flush in the L-brackets. So, I routed the L-bracket channels (rather poorly with a Dremel) in the tympan frame, and drilled holes in the chase frame so everything could sit flat. For tympan paper, I just used plain brown heavy paper I use for miscellaneous projects.
In lieu of gauge pins, I made some corners out of heavy paper. I'm thinking that for my purposes, some of those scrapbooker's photo corners would work well, since that's essentially what I made.
The end result feels a little hacked-together, but I'm impressed with how well it works. I didn't have any problems with alignment, and for deeper impressions, I can add packing to the back side of the tympan paper (I've found paper shop towels work really well).
Now, to find a place to store the press ...