Go back to your 16v AC setup, don't use the Honda reg rec.
16v on your meter is RMS, so you're actually peaking at about 21v AC. I had a similar issue to yours when I put a regulator on the AC side of the circuit and not the DC.
Let your unregulated AC flow into the rectifier (I use a bridge rectifier off the shelf from Radioshack rated 250v 50 amps or similar ), you can also build your own with 4 rated diodes of say 10a (HPI generates about 5A DC on the lighting coil).
Then send it from the rectifier to a filter/resevoir capacitor somewhere between 1000 and 4700 uf. I'm using 2200, but when I charge my phone my DC voltage sags from 12.7 to 11.4 so I might step it up.
After the cap, send it to a regulator,noff the shelf or otherwise. I built my own, but you can find a not rectifying regulator easy too.
And then boom, stable clean DC even at idle with no battery