You have to use a hood insert. Not sure you can purchase for that price................also here is the big issue. To recirculate, you have to vent it back to the room. How do you plan to do that?

https://www.amazon.com/

Okay... we will rethink. My apologies.

You can totally do this. You need an insert that has a ductless conversion kit option. Just a quick check and here is one by Broan. Obviously it is not ideal to recirculate but sometimes there is no choice

This one I linked recirculates right back out the same place. It is a little more money than you wanted to spend but maybe you can find it cheaper elsewhere

https://www.build.com/broan-pm390/s651824

Unless you actually cover the burners with the capture area, you might as well not have a hood at all. That picture isn't deep enough front to back to do that. Recirculation means having a grate open o the room for the air to come back into the room. Now, that could be in the room behind the kitchen instead of the kitchen, but air has to have an entrance and an exit to have flow. You also need a decent quality insert. You need to raise your budget. And study a few building codes regarding combustible surface clearances if you are going to build that yourself.

What kind of stove do you have?

I don't think recirculation will work with built-in/insert hoods. The air has to come out somewhere but where will it come out if the entire hood is covered all over? If you need to do a built-in, it will have to be ducted. You can check out our Premium 30" built-in hood which has different exhaust options for better flexibility:

Kobe Premium Built-In Range Hood, Stainless Steel, 30" · More Info