Tonight Beanie finally met the next-door neighbour's dog Salem and it went fine! I was out taking him for a walk around the block, and we ran into Salem and the neighbour out the front of a house around the corner. Beanie of course went feral, but I got the neighbour to do the thing that a dog trainer did YEARS ago where we picked both dogs up and let Beanie give the other dog a good sniff, then put them back down. After that there was a bit of dancing back and forth as they continued to sniff each other for a bit, but once they were over it that was pretty much that and they mostly just ignored each other while the neighbour and I chatted.
Very interested to see how Beanie goes next time we run into them, if he remembers Salem or not. It seems to be just the initial introduction to new dogs that he has a problem with, once he's had a sniff he's fine.