* The DOM isn't "slow" per se, but using the DOM for this kind of application is usually the wrong path to take. As soon as I need something even slightly off the well beaten path of web applications I tend to reach for canvas based solutions.
* Libraries like EFL are quite nice to work with but also quite low level. By default that means C, but there are bindings for other languages.
* Some of the things you describe (like rendering spectrographs) I'm not convinced are as CPU intensive as you think they are? Have you benchmarked? Winamp was doing these kind of visualizations back in the 90's. I kind of doubt if you really need to use shaders to render a typical DAW if you're smart with which parts need immediate mode rendering and which can use retained mode. Using render targets etc seems like a smart way to do a DAW, as most parts are actually quite static and only move around the screen.
* I have a feeling - just a feeling - that a lot of people choose Rust for projects because it's one of the newer shinier languages, but as you've discovered, maybe that's not necessarily smart. I would choose Rust for its concurrency paradigm - which might be useful for a DAW! - but for building a UI I would actually personally choose something JavaScript-based. The productivity is just so much better than most other languages, for building UI's. This is not just my experience - I worked for some time at a large public listed US company who used JavaScript for all their big UI projects on embedded hardware, what they changed over time was the underlying rendering engine, but they kept JS because 1) hiring devs 2) productivity.
Good luck, it's a super ambitious project, but I'm sure it's amazing fun to work on! :)