Welcome to the forum and to fly tying. I'd recommend a read of this. These guys did a very good job reviewing vises within your budget. Mind you it slightly dated, but for the most part very much relevant.
Fly Tying Vises In-Depth Review by Fly Fish Ohio!
Personally, I tie on a Peak. Very happy with the build and use. I've not found the head too bulky until I get down to say 18 and below. When I'm at that point, I use the Midge Jaws. I have tied thousands of flies on mine since 2009 and have not had one single issue.
You might also want to have a look at Regal Medallion or the Traveler series. The other to have a look at is the Renzetti Traveler. Both great companies with well respected products.
Good luck, let us know what you end up with.