I recommend the second option. Rasterize your font with stb_truetype to a bitmap, and draw quads yourself
It shouldn't matter that much if it takes a few frames to layout and/or update your ammo count or whatever.
Now, that's not to say it has to be that way; real engines have a separate render thread that would sort of solve that problem. And they run the game update code in parallel, and all kinds of other stuff. But, by the time you have all that stuff, you can probably rasterize a font and draw quads ;)
But don't take my word for it. Maybe it's better now. Make a thing that moves some divs around in a requestAnimationFrame callback and take a look at the profile. I bet 2% of the time the layout engine takes 30ms+ to do basically nothing
Also, you can render and rasterize fonts + dynamic text to a texture and UV map it to a quad in 3D space if you need to. This is pretty inexpensive and easy to do as well.
Lastly, the most difficult but performant option is using a shader to compute the font rendering. This is basically moving the font raster from CPU to GPU and there are various shader examples for this in GLSL (popular shader format used by WebGL/OpenGL) that you can use.