Welcome to the forum! He should be fine without the filter (only my baby tank had a filter). Bettas have a labyrinth organ which allows them to breathe air, so you don't really need a filter for him. Just a heads up with your 5 gallon tank (unfiltered) you should be doing a 50% and a 100% (or near too) every week. Once you get the filter you can drop down to two 50% water changes every week with a near 100% change every other week.
I do look forward to seeing this guy, he sounds gorgeous!
Feel free to message me any questions, I'll try to help the best I can!
Rummy- Mini Rex Rabbit
Diz jr- Betta
Yoshi- Crested Gecko