That's interesting...I wonder why they don't want you disconnecting the
battery?

The only thing I can guess is that there is something in the car that is
programmed and doesn't have a backup, and removing the battery clears it.
I have heard of this on other cars but can't remember what it
was...transmission pops into my head for some reason.

In order to replace the battery you have to 'hold up' the voltage with an
external source. At Toyota dealers, they have something like a battery
charger that clamps securely onto the battery clamps so the battery can be
removed. I would imagine a good battery charger would do the same thing,
but make sure the connections don't come off until the battery is replaced.

have no clue.. I changed mine with a new battery just because I could... no
problems...

Do you have the toyota warranty? you should be able to take it there to have
it fixed.

The dealership probably has a supplemental battery to maintain power while
the battery is being charged.

I'm not sure about the Yaris, but the usual reason to avoid disconnecting
the battery is that some systems have to be re-initialized, like sunroof,
navigation, theft deterrent system, etc.