Filter: any sponge filter rated for 5-10g should be sufficient. You need an air pump, air line tubing and check valve for it.

Example: https://www.amazon.com/Hygger-Aquarium-Single-Sponges-Submersible/dp/B07XZ625DG/

https://www.amazon.com/UPETTOOLS-Aquarium-Biochemical-Sponge-Filter/dp/B08JGQ1ML5/

Air pump kit example (attach to sponge filter instead of air stone): https://www.amazon.com/FEDOUR-Aquarium-Oxygen-Bubbler-Gallon/dp/B0BVLH4MKG/

Also!!! PLEASE DON’T RUSH TO GET THE BETTA! A new aquarium and new filter needs 2 weeks-1 month or so to cycle, for beneficial bacteria to get properly established. https://injaf.org/articles-guides/beginners-guides/the-nitrogen-cycle-and-the-fishless-cycle-getting-your-aquarium-ready-for-fish/

Get Seachem Stability if you want to help speed things along (and double/triple the dosage as you like - more beneficial bacteria added will help speed the nitrogen cycle more, and any extra will balance out and die down eventually).

Mini Sponge filter (corner) - https://www.amazon.com/HIKARI-Bacto-Surge-Foam-Filter-Mini/dp/B00TTBLDXY/

Air pump kit (includes all necessary accessories including check valve) - https://www.amazon.com/FEDOUR-Aquairum-Outlets-Aquarium-Bubbler/dp/B0BVLH4MKG/