Quoted battery life for the Marshall Kilburn is "up to 18 hours". For busking in open spaces, the volume required is going to considerably more than that needed for indoor private use, and high volumes will consume lots of current from the battery, even though it's a class D amplifier in there. In fact, run flat out, the Kilburn will probably only deliver a couple of hours usage playing music, although drum sounds, by their percussive nature, will probably give three to four hours run flat out.
The speaker is physically relatively small, giving limited bass response. The low frequency response rolls off around 60Hz, according to the specification, so some of the deep sounding drum sounds may not have quite the punch that might be heard through a decent pair of headphones, which can have a low frequency roll off at 20Hz or below.
Also, the Kilburn is a mono speaker, so some of the "acoustic" qualities of the wavedrum that rely on stereo separation, will be lost or muted.
Having said that, there are some quite enthusiastic reviews on Amazon.
It is also worth trying the speaker out before buying. Get the correct cable first, and bring it to the store with you.
Here's a useful link to a fairly comprehensive summary of a good many speakers useful for busking. I think they are all battery powered. Probably best to avoid the ones that run off 8 x AA batteries. They start to get a little expensive to run:
The Behringer MPA40BT-PRO Portable PA System seems like a good bet for the money.
The AER Mobile Compact 60 sounds very good, but is very expensive.
The Crate TX50DB finds favour amongst some of the busking fraternity.
If you have the time, I suggest checking out all of those in the link, if possible, and taking a good look at any independent reviews. I would suggest avoiding the Ion brand. Very thin audio, in my experience.
FWIW (this post includes mention of gear I haven't used) --
I've used my Wavedrum with an EV ZXA1 powered monitor, 8" woofer, and it sounds pretty good. That speaker takes AC power, and a lot of it.
I suspect that anything less than an 8" woofer isn't going to do well with the WD's low-pitched drum sounds. And there's not much power behind the Marshall amp/speaker mentioned previously. I wouldn't use Bluetooth; I believe there are latency problems, and even a little latency is a bad thing for a drummer.
THere's a Pignose "Hog 30" unit, 8" woofer, 30 watt amp, 28 lbs, designed for keyboard and bass players. It has a rechargeable battery built in, good for "hours of playing". There's no separate tweeter. It might be worth looking at.
If you can use a shopping cart, it's practical to wheel around one or two deep-cycle 12-volt (or two 6-volt) batteries, and a small DC-to-AC inverter. That lets you use _any_ small amplified speaker or PA system.
PS -- EDIT -- the weight of the "Hog 30" is 16 pounds. 28 lbs is the shipping weight.