The output from the BeatBuddy is plenty loud for ‘industry standard’ for both the headphone output and the pedal output so the getting a good signal out of the pedal isn’t the issue unless you have a bad pedal. You need to choose a speaker designed for one of those signal levels.
A good friend uses a small Bose speaker fed by his iPhone to provide accompaniment when he plays clarinet in small public settings. He uses the headphone jack to feed the Bose. There are plenty of other small powered speaker options that work with headphone-level output
If you want to play with a real band, then you need to look into a power PA speaker or keyboard amp in the $200-$300 range or more. At the low end, get a used Peavey KB300 or an Alto 12" or 15" powered speaker - they have a good reputation and are easy on the budget if you don’t want to buy used. These types of speakers are designed to work with the normal pedal output. Get two if you want stereo but you should be just fine working in mono with a single speaker.