I'll add a related real life debugging tip I've seen play out at least twice in the real world:

If you can't get an alarm to shut-off or stop beeping, no matter what you try, then give serious thought to the possibility that there's a second alarm nearby which is the source of the noise.

About a year after I moved into my previous house, we heard that familiar chirping sound.

After 3 goddamn hours of searching, and having run to the store to buy 9-volts and replaced every goddamn battery in the house, we found a carbon monoxide alarm thrown INSIDE the return air duct of the HVAC unit...

Which is a reasonable place for one, as it will trigger from excess CO anywhere in the house when the HVAC is running. The problem is only obvious when you're not the original tenant who installed the thing.

Wonder if a full battery with a resistor in series would prolong the agony.

Or a NiCd in place of an Alkaline battery.

Here is $5. You are right.