Okay, so I did it: I tested what I was thinking about, and, as a result of the testing, I rewired my own doorbell chime.
TL;DR:
I discovered that the red house wire attached to my doorbell chime’s TRANS terminal was actually coming from the doorbell, and the red house wire I’ve had connected to the Chime Controller’s black wire (this red wire which was originally connected to the doorbell chime’s FRONT terminal) was actually coming from the transformer. These had apparently been reversed at the time of the old “dumb” doorbell’s original installation, so I swapped them to their proper locations today.
More Detail:
I installed my own Video Doorbell v2 roughly 11 months ago using the standard wiring installation for the Chime Controller, and everything worked as expected from that time until a couple of months ago, when the chime stopped ringing with a button press. I tried a number of the typical troubleshooting steps and eventually got the chime ringing again somewhat reliably by using the alternate wiring instructions, but the ding-dong! is often accompanied by an unpleasant rattle/hum sound. Wyze Support says that “the buzzing/clicking sound on the chime is normal when an alternative wiring method is used”, but this seems odd to me, and trying to think my way through @fabron72’s wiring situation got me to explore and reevaluate my own doorbell wiring situation, so now I have a better understanding of what’s going on.
This is what I did:
- Tested the voltage at my 10VA/16V transformer and read 18.5 V AC.
- Turned off power to my doorbell.
- Disconnected the house wires in my chime box and labeled the red wire (what I believe would equate to one of @fabron72’s black wires) that was attached to the TRANS terminal. (I wrote “Transformer” on a piece of tape to differentiate it from the other red wire.)
- Untwisted the white wires in the chime box that have been twisted together and capped with electrical tape.
- Clipped a test lead to the red and white wires at my transformer, effectively shorting across them (probably unnecessary).
- Attached my multimeter to the red “Transformer” wire and its paired white wire (as they come into the box through the same insulation sheath), put it in resistance measurement mode, and turned it on, where it settled on “1”, even after double-checking all my connections. Oops.
- Attached the multimeter to the other red wire (the one that was initially on the FRONT terminal and has instead been twisted together with the black wire from the Chime Controller) and its paired white wire and tested resistance to get a realistic reading of conductor resistance (which I didn’t record). This is the actual wire pair coming from the transformer!
- Removed the Video Doorbell v2 from the wall and disconnected the wires so that I could test resistance again with that air gap and with a test lead clipped across the wires in order to confirm that the red wire which has been connected to TRANS this whole time is actually coming from the doorbell-side wire pair.
- Reattached the multimeter to the actual transformer wire pair (now that I know which it is), put it in AC voltage measurement mode, restored power to the doorbell, and turned on the multimeter, which read 18.9 V, confirming that this is actually the transformer wire pair.
- Turned off doorbell power again.
- Reattached and remounted Video Doorbell v2 outside.
- Rewired my chime box:
- Twisted the two white wires back together, taped the exposed conductors to insulate them, and tucked them out of the way.
- Attached my known transformer red wire to the TRANS terminal along with the red wire from the Chime Controller.
- Attached the white Chime Controller wire to the FRONT terminal.
- Twisted the black Chime Controller wire together with the remaining red wire (the false “Transformer” wire that I now know is actually coming from the doorbell) after removing my temporary tape label.
- Restored power to the doorbell, waited for it to boot up and load a video stream to the Wyze app, and then tested by pressing the button to get a nice ding-dong! from the house’s built-in chime now that the wiring is correct and the Chime Controller is wired back with its true standard wiring installation.
So far I’ve tested the button only twice since doing this, but it’s worked both times, and I’m optimistic. Now I can report my results to another topic where a user described changing wires at the chime. I didn’t get an answer when I requested follow-up there, but it feels good knowing that I’ve been able to go through this process and figure it out myself, learning some things along the way. If this isn’t a lasting solution for me, then I’ll update this with new information later.
I still feel like I owe @fabron72 thanks for bringing this problem here, because I probably wouldn’t have thought my way through the wiring situation otherwise.
I guess in a way you still are, though maybe not in the manner you expected! 
Having noodled my way through this, I still think I’m right about your wiring and believe that you could resolve your issue without engaging an electrician, especially if you had a working multimeter. I also suspect that you could’ve gotten everything working as expected with your original transformer, but that is maybe less relevant at this point.
I’m interested to learn how things work out on your end.