Vic, this is an interesting poser. My first reaction is that both are most definitely Kundos, based on the fact that Kern & Söhne did not exist as a company until 1937, and to the best of my knowledge only made movements on the Kienzle pattern. That came about because Kern and Link took over the Kienzle 400-Day business in 1929, then Link left in 1932 and Kern continued as Kern Uhrenfabrik until 1937 when it became Kern und Söhne.
The catalog you have is Kern & Söhne, thus could not have been printed prior to 1937. The pendulum shown on the Model 501 does appear to be similar to the known Kundo No. 37, but we don't know what is the movement layout. If that were available it could settle the question.
Next, all the Kern standard size movements made after WWII continued the identical layout, train count, pendulum design, etc. of the original Kienzle clocks. This was changed only when the narrow plate designs were introduced.
Finally, no known documented Kern movement has had a serial number. One thing we don't know for sure either is whether K&S used the logo that was registered in 1937 to stamp on their clocks . . unfortunately with no serial numbers it is impossible to tell whether a K&S logo clock was made before or after WWII, especially considering that the only pendulum seen with them is the Kienzle design 4-Ball pendulum.
Before reaching a conclusion however, I need to do a search of my data and photo archives to see what I can find to support one direction or the other. Without doing that I can state the following with certainty:
There are dozens of Kundo clocks in my data that have (a) no suspension guard, (b) the tubular suspension guard, and (c) the Kundo square section suspension guard. Among these are MANY that have the Kieninger & Obergfell or KO logo, on all three types. There was no suspension guard on initial production in 1923. In 1924 the tubular guard was introduced and used until 1932. In 1926 the square form Kundo guard was introduced and continued until they ceased production in the 1980's. This guard was the only one used after 1932. From 1926 to 1932 clocks were produced with all three versions.
There are also many differences between various Kundo movements, for example:
Some have an eccentric on the front plate to adjust escape wheel depthing.
Some have different size mainspring barrels . . one with 25 mm from the center of the mainspring barrel arbor to the center of the first wheel, another with 26 mm for the same measurement.
The click wheel bridge is a stamped single piece on many (you call it the square edge bridge) and the 2-piece figure 8 shape on many others. The majority have the latter type.
I had not checked the shape of the cutouts in the wheels, but will do so and report back on that. I would not be surprised to find both types on clocks bearing the Kundo logos. I also had not observed differences in the serial number stamping but will check that also.
It may be a couple of days before I get back with the info.