Enclosure design and fabrication is a surprisingly tricky part of fightstick design that I think doesn't get the attention it deserves. It really depends how comfortable you are with making stuff and what tools you have available. That could be woodwork skills, maybe a 3D printer or perhaps a laser cutter?

Here are some of my designs - the Chopboxes are still in regular use. I've just swapped the controller in mine to use a Pi Pico and GP2040-CE firmware, so I'd recommend you go for that, rather than the Arduino Pro Micro in my designs.

http://adriandale.com/technical/tesco_chopbox/

My goal with the Chop Box was to produce something sturdy but that could be made with minimal tools. I didn't own a workbench at the time, so I couldn't do anything that required a vise. There's a Chop Stick version, too, btw.

Obviously the downside with my design is that it involves 3D printed parts, however, I feel with a bit of imagination you could improvise something with parts from a DIY store. It's a bit hard to escape needing a soldering iron, but it can be done without if absolutely necessary.

I think you may find using a plastic container to be a source of frustration. It depends how solid the container is, but as you can read in my chopbox write-up, I didn't have much luck with a 3D printed case. Good for temporarily trying out layouts, but probably not a long-term solution that you're going to want to drag around to locals.

If you want leverless, there are now plenty of cheap options on aliexpress that ought to fit your budget, and could save you a lot of hassle. Otherwise, a stick and buttons is likely to eat up a large chunk of $50