Minor backfire after jet change?

Not sure what the reed valves are. There are two small covers on each side of the top of the head that had ports for smog vacuum hoses. I did remove these two smog vacuum lines and capped the ports off. Are the reed valves under these covers? And if it's a problem, what can I do about it?

Yes under those covers are the reed valve petals, these were part of the Air Injection to the exhaust for emissions. Again if hooked up and working correctly it will not cause any loss of performance. If they are bent and leaking, then air can go into the exhaust when un-intended. So if you have the bigger say 1/2" hose fitting on the top cap plugged off, then this is probably not an issue. Not sure if 1 cylinder could still get air from its joining cylinder if plugged but reed valve petal leaking. It might be worth a look.
I think you can get needle shims from most bike shops, hardware washers are probably to thick and would vairy in thickness as they are not shims but washers. The other would be either buy a richer needle or needle jet. I saw where you are drilling after soldering, this can get you in the ballpark, then you can get some jets in the correct and even sizes.

Sounds like progress.
I have the same carbs and needles and I dont think you can put a washer under them.. not totally sure on this.

If it accelerates fine and idles fine then it sounds like your drilling the jets is probably ok. I used 62/140 and 2.5 turns out without problems before moving to 70/140. If you need to buy Keihin jets then Z1Enterprises have them for about $2 each.

In adition to the suggestions above on exhaust leaks and reed valves - which we never had up here in Canada - perhaps take a look at how the pods are attached to the carbs and the carbs to the motor. Earlier this season I had some popping in the exhaust. May or may not have been caused by poor fitting pods but when I got new hose clips and paid carefull attention to how I attached the pods my popping dissappeared.

Good luck. You sound so close to having a perfectly running machine

PS Also thinking drilled out muffler