You can fly in Class B, C, D airspace with a Sport Pilot license if you’ve received and logged the proper training from your flight instructor. See: https://www.aopa.org/advocacy/advocacy-briefs/frequently-asked-questions-about-sport-pilot
I’m using a YAMAHA SHINDENGEN FH020AA regulator. It’s designed for three phase stators but works fine with our single phase types. It uses MOSFET technology and wastes less power in heat than the traditional types used with rotax engines. It maximizes the output from the 165 watt coil on a 582.
That regulator requires some load on it (1 amp minimum?) to regulate, besides the filter cap. Make sure to keep something on when the engine is running. Maybe lights of some sort. Radios or other avionics aren’t a good choice because they might not like voltage excursions when starting up or shutting down. Or, go ahead and upgrade to a better regulator. Lot’s of people like the Key West. I use a motorcycle regulator.
Check to see you’re getting 12-14 volts at the power supply and the wiring is good out to the stobes. I had mine go out and found an internal fuse in the power supply was blown. It’s a pain to open one of those up because they used rivets which have to be drilled out (OK, they’re aluminum so not too challenging). If you do open it, remember you’re dealing with high voltage. At Oshkosh one year, I found Whelan was selling refurbed units they pulled from under the counter. Wouldn’t hurt to ask them if you still need one by then. Or go LED.
You’re going to have to do some fabric work. With the cut fabric, the integrity of the covering is compromised and it will not be airworthy unless you terminate / patch it properly even if you are adding additional access covers. This would be a great time to call in some help from a local tube and fabric expert to get you on the right track. The work you’ve already been doing is more advanced than fabric skills so you shouldn’t have any trouble picking it up. Thanks for sharing your fabrication adventures.