The card is (unsurprisingly) an ISA slot, an Intel EtherExpress 16. I forget exactly where I was with it when last I poked at this earlier this year, but the promising thing is I have it worked in MS-DOS 6.22 without issue. There are some utilities from Intel that I found that write EEPROM on the card to setup its IRQ and all of that, which is how I got it working in DOS.
Perhaps I’ll take another swing at it and document more precisely what the issues are and what I’ve tried so far. I wrote up the acquisition, repair, and resuscitation of the machine on a series of posts on my blog under “Project 386” beginning with https://justinmiller.io/posts/2020/04/26/project-386-part-1/ so I have intentions of adding some more posts.