Everything polar bear said (quelle surprise!). You actually have the perfect experimental set up: the current problems are clear, so there's a good opportunity to some specific before/ after. The teacher doesn't think accommodations are needed because that's not the problem? Well, let's test that theory. Pick one or two specific common problem situations and state, very explicitly, what accommodations will be used to address them for the next week or two. And prove/ document that the accommodations were *actually provided*. Then look for a difference.
Now obviously, a kid who's been struggling with LDs in a hostile environment isn't going to miraculously perform the moment some accommodations kick in. But if you pick your situations carefully, you will almost certainly be able to see some real differences. Ideally, work with your much-more-supportive administration to get an *external* person for a week providing the accommodations - say, scribing those "show your work" assignments that have proven so difficult.
I'm pretty sure a scribe/ EA for an hour a day would show some pretty notable differences in what your son is producing during those time periods. You'll never get this teacher on your side, but you may be able to reduce the hostility to accommodations for the rest of the school year.