What is the elevation where you live and ride?

With a stock paper air filter and DYNA muffler - a #150 main is normally plenty of fuel.  

Did you clean your carb?  What is the condition of the TEV diaphragm....if it is old and stiff it may not be working (The TEV is a Transient Enrichment Valve, and it provide more fuel during high vacuum events (like when you close the throttle and create a high vacuum).

When you close your throttle abruptly you create a high vacuum and the slide has dropped and closed off most of the fuel flow - this creates a lean condition and the fuel/air mixture that cannot be ignited by the spark plug....but it can be ignited in the hot exhaust header.  This creates the noise you are experiencing.

You can minimize the noise by avoiding a completely closed throttle while slowing down.  Open up the throttle just enough that the noise stops, and don't completely close the throttle as you shift.....leave it open just enough to prevent the noise.  This is a big single with a carb, and it is not as refined as a fuel injected engine......you can adapt your riding style to work with the limitations of the old girl!

I firmly believe you should jet your carb for how it runs while "ON" the throttle - not for what happens when the throttle is closed.

The slide needle is what controls the mixture during very low throttle openings.  Three washers is the normal number to use.  If you feel a slight hesitation or uneven surge while at very light throttle openings up to about 1/4 throttle - then remove a washer and see if the smoothness improves.

An engine that is running too rich will run - but it will run much better with a proper fuel mixture.  A properly jetted engine will have a slight surge and lean feel while the engine is cold - then run nicely once the engine is warmed up.  If your engine takes a few minutes to start running smoothly - that is a good indication the bike is not jetted to rich.

A spring goes under the needle clip, and it pushes the needle upwards.

The white spacer goes on top of the needle and pushes down on the spring and establishes the height of the needle.

The white spacer thickness pushes the needle down far enough that the mixture is lean and good for pollution control....but not good for smooth running.

You can either cut down the white spacer height - or replace it with 3 washers.  This allows the needle to be a little higher in the slide - which provides a richer mixture.  The more washers the leaner the mixture and the fewer washers the richer the mixture.  Evey carb I have ever done was happy with a #150 main, #50 or #52.5 pilot and 3 washers.....with the exception of 1 carb that had a surge at light throttle and removing 1 washer cured that.  My riding elevation is 525-800' msl.

Check for an air leak at the carb inlet - sometimes the Oring can be bad or the plastic spigot can be cracked.  Take an unlit propane torch and while the engine is idling see if the idle speed changes when you run the unlit propane around the carb body and inlet spigot.

NOTE:  The washers need to be steel - the brass washers have too big of an OD and will not fit down into the slide.  It is often hard to find the steel washes at a hardware store - for some reason the people that stock the nuts/bolts supply the store with brass washers.  You can also buy specific washers for use on carb needles from folks that sell Mikuni carb parts.

Here is where you can get a TEV valve to replace your 19 year old one.
https://www.ebay.com/itm/306128009612?_skw=ls650+tev+valve&itmmeta=01KZ68HT06...

The community’s previous responses have been very helpful, and I would like to contribute my own experience to the discussion.

I have found that overly rich jetting can paradoxically cause a lean condition during deceleration, and an idle set too low often exacerbates this. Frequently, the transition from a rich mixture to a lean one during deceleration is more problematic than running lean across the entire power band.

My advice is to ensure your idle is set to approximately 1150 RPM and keep your jetting close to factory specifications. Dave’s recommendations provide a solid, near-stock baseline; if you are still experiencing issues after these adjustments, you likely have deeper mechanical problems. Remember that you should be jetting for performance rather than using it to mask other issues. If you find yourself chasing carburetor settings to hide poor running conditions, it is likely that you have underlying issues elsewhere.

Edit: For reference, I am currently running a 45 or 47.5 pilot jet and a 145 or 147.5 main at 500-1100 feet of elevation.

Make sure you don't have any air leak between the header and the head
That will give you mad afterfires
I have two crush gaskets in there

Smart move... you don't want to do that
If you hold a cigarette or some sage, whatever near the gasket you can see a leak, but you're right... don't strip that booger

You're doin' all the right things
Good job

Ok status update.  
I ordered a new TEV as suggested (thank you!), which took a while to arrive, and installed it.  Yes, the rubber on the old one was not soft and the valve itself was somewhat stuck in the bore.  I pulled it out and there was some discoloration along the annular sealing surface.  I sprayed out the bore and installed the new one and went for a ride.  I was hopeful but . . . no change . . . BANG at lift off!
I removed the pilot jet and installed a 55 pilot jet, which I think should be overkill.  I was hopeful but . . . no change . . . BANG at lift off!
I took an all day motorcycle safety techniques class.  During the 30 minute ride there and back, overall, the S40 ran well at all speeds (up to 65 mph) . . . but banging all the way at liftoff.  If anything, it may be running a bit rich as it seemed to feel a bit boggy but I am not sure?!  

So current config. is stock except Dyna muffler, with 55/152.5 jets, 2 steel washers instead of the plastic piece.
So, my next plan, in order is as follows BUT I would like your feedback:

1. Double check for leaks at the carb and exhaust as suggested above.  I just bought a propane torch for checking for intake leak.
2. I just bought a KN filter.  Try that and see if it feels a bit better which would seemingly indicate that it was actually running rich.  But I don't think this will change the core annoyance. But I will assume if banging gets worse, then running more lean because of the KN? I will use whichever filter that makes it run better, if I can tell of any difference.
3. Presuming it is still running well, without much difference, take the top off the carb and remove one washer, which just leaves one washer.  
4. If that does not help, I have read some people clipping one loop off the TEV spring to make it weaker as helping the backfire situation.  Should I consider this??  The main concern is that I would have to find and buy a new spring if it does not help.

Other options if none of the above works is to:
5a. Just accept the constant backfire, trying to minimize with slower throttle let off ... but I don't think I will be able to accept this.
5b. Just ride the rest of this season with the backfire, and during the winter, take the carb apart and install a full rebuild kit myself or have someone else do it for me.
5c. Give up and buy a replacement carb from Murray's Carbs.- with confirmation from Murray and/or from here that their carb will get rid of the backfire problem. But they are also very expensive. Anyone have experiences with them?

Sorry for the long post but I would appreciate any further thoughts you may have.
Thanks,
Daniel
 

Thanks SoC.  Sorry I did not know there were different terms for the popping.  I am pretty sure that afterfire is what I am dealing with.  Have not had time to look at this but my last ride, I did count and it is generally two pops on upshift and two or three pops on throttle lift so it is at least double yours.  I also noticed after turning off the engine, there is a small poof from the exhaust now, which is something new.  I guess I am running a bit rich.  Anyways, I hope to play around with it some this week.