Around 10 or 15 years ago I spent a fair amount of time messing around with circuits like this.
I think this “Little Gem” is a variant on the “Lil Smokey” battery amps that were initially sold stuffed into cigarette packs, before the company switched over to selling them in cig pack sized plastic cases. Remember those? One schematic:
http://farm5.static.flickr.com/4032/4267267797_792fb8ee10_o.gifThat amp used an LM386 chip or variant, and yes, a 9-volt battery, to put out something in the half- to one-watt range through a little plastic (mylar?) speaker housed in the same tiny package. Cheesy, but gritty and fun - enough to scare the folkies at a campfire with some Sabbath riffs, anyway.
The plastic case version had a speaker jack, and most reviews you found on the web of those Smokey amps included a line or two about how awesome and surprisingly loud that little 1-watt amp sounded through a 4x12 speaker cabinet.
I think in your case, there is wisdom to be found there.
If you play a 1 watt amp through a highly efficient 12 inch speaker, and can tolerate some amount of overdrive, or even something similar to “soft clipping” present in most pop guitar sounds, you might have something loud enough for busking. I cannot emphasize this enough: an efficient speaker makes a really, really big difference in how loud an amp sounds. A 5 watt amp through a small, inefficient speaker can absolutely sound much quieter than a 1 watt amp through an efficient speaker. This makes sense to people when you say it, but then they forget it and focus only on how many watts the amp is rated for. That’s a mistake if you’re going for volume.
Making a 12-volt pack out of eight AA batteries will let you run an LM386 based amp notably longer than a single 9 volt battery, and will give you a little more volume and headroom - maybe even enough more to notice. You can buy an eight-pack battery holder for AA batteries - check on Amazon or use your google machine for options.
My advice, for what it’s worth: If you must have something in the 10-20 watt range, or you must have crystal-clear clean tones, you should buy a ready-made battery amp. They are available for purchase. But if you want to build something and are ok with some grit in your tones, build that Little Gem and try it through an efficient 12” guitar speaker to see if it seems loud enough for your purposes.