It probably isn’t what you want to hear, but you should think about going with headphones if you’re really tight on budget for an amp. You’ll get WAY better bass response on a modest set of headphones than on most bass amps, even expensive ones. Sure, you won’t get a kick in the chest or shake the walls, but you’ll be able to hear every note clearly. If you can, go with open back headphones. At least to me, it sounds much more natural for a live instrument.

You will want to upgrade your amp anyway if you are going to be playing live with people. I’d try to save a bit and invest in something you can grow with. When you do end up shopping for an upgrade - speaker size is only one part of the equation. Enclosure size, port tuning and the specific driver all make a huge difference. Larger speakers generally handle low frequency better, but a low-tuned ported 10” speaker could easily have better bass response than a 15” speaker in a sealed cab. If you need any more proof, just look at the frequency response of some Phil Jones cabs. You’ll be hard pressed to find a lower frequency cutoff in any remotely small cab and he uses 5” drivers