We've added a DIY autofill solution which seems to be doing the trick.
The pond holds less than 180 gallons (according to the water meter attached to my spigot) and it's quite shallow. Letting the pond level fluctuate naturally wouldn't make for very happy fish, so I wanted to find a way to maintain a constant level despite our very slow leak.
We tried to fit a skimmer into the pond, but it's just too small, so none of the ugly plastic autofill devices were an option for me. This thing was going to be visible, front and center. It had to look like I meant it to be there. Enter one Amazon used "Acceptable" brass ballcock and valve assembly, a clear plastic warehouse store chocolate almond container filled with river rocks and some tubing we ran up to the rain barrel. The previous owner of the house was an artist welder, so we've got many funky metal sculptures on the property. This thing fits right in, and it works! It took a couple of tries to find the right height adjustment, but it was easy enough to do. It sits on the bottom of the pond on one layer of leftover mosaic tile from a bathroom remodel and I happened to stumble across a hollowed out rock that fit perfectly over the plastic container's lid (though the black lid was hardly visible).
The pond level remains constant. We can see the level of the rain barrel going down, but not enough to empty it before the next rain. I haven't measured, but I think we lose about 1/4" in the rain barrel each day. All the creatures seem very happy. Our water test results are all within norms. I'm watching the copper levels now, but so far no change from our tap water. We have somewhat alkaline water which makes it less likely that copper will leach from the brass float. If it becomes a problem, I'll coat the float with a pond safe resin to encapsulate it.
I wasn't sure we'd be able to get the pond going without a big investment in time and money, but we managed and I couldn't be happier.
We don't seem to have had a problem with new pond syndrome because this pond had been filled with rainwater and muck for years and had achieved some kind of natural equilibrium. Despite having repeatedly emptied and refilled the pond I think some of the good bacteria survived. We also have a barley straw ball tucked in the waterfall (to inhibit algae growth) and when we noticed some bubbles starting to down below the waterfall we added a couple of Pond-zyme treatments according to the directions.
I hope this will encourage people who end up with a "bonus pond" to give the rehab process a try and not to get discouraged right away if there's a problem. Get creative and you might find a solution you love!