Music!

I'm not sure if this is something you'd be interested in, but one of my close friends had a DJ for the first half of their reception so they could accommodate slow songs and songs from other decades for the older crowd (who typically leave earlier in the night), then had a live rock band for the second half of the evening for the younger crowd. It was definitely a big hit and kept the reception feeling upbeat!

Not sure where you are located but check out https://www.bachelorboysband.com/live-showcases? They offer a quality, 5-piece band. This the smallest size wedding band I've seen and they've played at our venue many times over the years. Clients are always happy with them. The other option is to go with a hybrid DJ/Live combo. I work with a few DJ companies that offer this. They will provide live music for ceremony and cocktail hour, DJ for the reception and then bring in some live musicians (usually 2) to accompany the DJ. I've seen everything from electric violins to bongos and saxophones. It may sound strange but it really is incredible!

Your request is stretching the idea of a "band" when you want 3 musicians. You might want to allow one or two musicians if the price remains close to your budget. With three, you probably would want a keyboardist who also can have pre-recorded drum/percussion tracks and then a fiddler or horn player (sax or trumpet). A keyboardist might be able to do some of what a DJ does or maybe a DJ can play tracks that a saxplayer or singer can perform to.

You can try putting a request for musicians at sites like the gigsalad website-- I don't know how well this works but I think you put in requests and musicians may offer bids or request more info. I just found this when searching about a friend who is a musician.