Audience familiarity with the Tommy/Birmingham pairing is the most likely explanation, and the number two son/number three son confusion was probably just an overlooked error.
Your off-screen death explanation actually adds depth to the characters. It made me think of a true-life story about the young Katherine Hepburn, who assumed her beloved teenage brother's birth date after he committed suicide.
My own favorite theory, as of two minutes ago, is a multiverse of many earths (courtesy of TV's The Flash) -- with each change of actor playing Charlie Chan we're actually looking at a different earth, similar to but significantly different from the earth that came before, so on Roland Winter's Charlie Chan earth there never was a number two son named Jimmy, just a number two son named Tommy who looks like Jimmy from Sidney Toler's Charlie Chan earth...
Now I'm just being ridiculous. Thanks again for the reply.
As I, too, am familiar with the "multiverse" as seen (hugely!) in "The Flash" TV show, your suggestion in that direction is a fun take. I was not aware of the Katherine Hepburn angle. Thank you for sharing that!
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My personal theory is that the actor Sen Yung had a bond with Toler and only wanted to be "Jimmy" for Toler. When Toler passed he retired the Jimmy character and became Tommy perhaps retaining #2 status as a compromise.
The Chan films were very popular and even though Monogram was a poverty studio they still employed good writers. I can't believe it would have been an overlooked accident to change character names without reason. After all, Yung was "Jimmy" in the last batch of Toler films. There would have been a whole slew of people who would have noticed the change before cameras started rolling.
Yours makes sense as a possible answer. I agree that the name change was certainly not an "accident" as Victor Sen Yung probably asked the big question regarding this when he first read the script for what would in the end be titled "The Chinese Ring" (working titles: "The Mandarin's Secret" and "The Red Hornet"), the first Roland Winters Chan picture. In the end, calling Jimmy "Tommy" was probably an executive decision.
