I've seen engines as small as B&S 3.5hp vertical-shaft with a 6V electric starter and simple 1-1.5A charging coil. Probably too small for your needs. Almost all of the older 5hp & up verticals with electric start will have a stator underneath the flywheel. If you use a Briggs engine, you can download their alternator chart which shows which stator was available for which engine(s). In many cases, more than one stator would fit a particular engine series, depending on the amount of electrical load you would have and the desired charge rate for the battery.
I had an 11hp horizontal snow thrower engine that had a 1.5A DC stator to power a headlight on the 'thrower. When I swapped the 120VAC electric starter for a 12VDC starter, I put in a bigger stator to not only power the light, but now also charge the battery. In configuration terms, this was by far, the cleanest, because after all was said and done, there was no visible change in the appearance to the engine.
I use my B&S engine as an example, simply because they have, by far, the best customer support over anyone else, both behind the counter and on the web.
I hope this opens up some possibilities not previously considered.
For my go cart project, I'm using a Briggs 18hp opposed twin with a 20A OEM stator and regulator combo I found on eBay to charge the battery, run a few lights, and possibly a radio.