After saying I didn't want to throw money at this problem, I threw money at it. I checked out some other VPS hosts and none of them offered anything substantially different than my host, at least not at the price I'm paying. So I paid to double the RAM for my server on my current host, then fired up my origami site again last night and it hasn't given me problems yet. Perhaps 368MB was just not enough to run SBCL and Apache and a bunch of Mongrel servers all together. Intuitively I would've though it'd be enough, but such is life.
I'm still keeping my fingers crossed, usually it took a day or two for SBCL to start gasping for life so there's still time for all my problems to return. I hope this works out, because I really liked writing my site in Common Lisp. Thanks to all who posted their own brief success stories and suggestions, telling me I was wrong to give up on SBCL.
I'm not a great programmer, that much I recognize and admit freely. If anything, I'm an enthusiastic programmer. I enjoy doing it, I enjoy my little successes, I enjoy learning from my frequent failures. I'm grateful for anything that helps me figure out what the heck I'm doing. Failing is a good way to learn.