Short of going full Solr, I found Relevanssi and decided to give it a try. It turned out to be actually really well made and satisfies most of my desires in improving search, including search result highlights.
Unfortunately, the speed of search suffers greatly. Regular WP search pages are generated in about 1-1.5s on my site (4400 articles or so, lots of tags). Relevanssi does 5-6s, and sometimes over 20s. Unfortunately, that's not good enough, but without going to full Solr I'm not sure can be improved. I'd welcome feedback, however, especially from the authors.