KaTeX works great in a headless environment. You can even do server side rendering if you want a browserless environment which can serve up straight HTML with no JS.
Both KaTeX and MathJax's biggest slowdown is the font loading. However, since KaTeX's CSS can be included manually (instead of letting MathJax put it in the DOM after load) it should load a little faster.
Also, looks like your use cases are similar to ours at Khan Academy! We've been struggling with MathJax on slow computers and iPads for a while, which is why we made this. Let me know if you want more info!