Alarm set off by small key

My guess is: That is not something you can "fix". It is meant to work that way.........so that opening it with "brute force" applied to the mechanical lock will create an alarm.

I think the alarm is really triggered when the door is opened, right ?
And not when the key is turned ??

 

Has the door lock actuator ever been replaced that you know of?

You might want to compare this experience (mutatis mutandis, it was in a Gen 1).

The lock actuator contained a switch to detect turning of the physical key cylinder, so it could tell the alarm ECU "it's ok, being opened with the key, not forced". The $17 ebay knockoff replacement actuator in my case flat-out had that switch missing inside. I don't know if the manufacturer was just using ebay as the outlet for their QA rejects ("ok, that one got built right, send to Toyota, that one got parts left out, sell on ebay") or it was just a different, really sloppy knockoff manufacturer, or what.

(Edit: looks like I misremembered which of the little switches inside was missing in mine, but the details are in the linked post. Your symptoms sound more like the other switch, anyway.)

Anyway, I think it was a pretty successful operation for the ebay sellers, because people would buy the actuators and test that the door locks/unlocks and call it good, and probably not run into what doesn't work until way later (how often do you open the door with the mechanical key?), by which point they won't be thinking it could be related to the lock actuator that was changed a long time ago, they'll just get annoyed with their car for randomly failing.

For the few customers that actually connected the dots and realized the actuator was bad, the seller would just send $17 back and call it cost of doing business. Ebay's policies are weird that way; if one small time seller sends you something broken there's a way to report that, but if there's a seller with an ongoing business model of just selling broken stuff and refunding whoever notices and complains, there's not even a way to report that so it makes any sense.

If it isn't that, of course it could be the other things that story would make you think of ... is the linkage from the key cylinder to that sensor securely attached? What's the condition of the wiring connector to the actuator, especially on the pins that carry that signal?