Would a 110 vac relay be activated by a 12-15V DC transformer?
And this would mean both the relay and the siren would be powered by the
same power supply (the dc transformer) except that the relay would also
switch the siren. Is this correct?

I WAS trying to be really simple which is why the light bulb or resistor
idea sounds good.

How much damage could I do by using a resistor box and starting with a
high value and slowly working down until it worked acceptably? I guess I
could damage something in the process, but how likely do you think it
would be?

Joy



He means connect the relay to the same power source that the transformer is
connected to, and use the contacts to switch the output of the transformer.
That route occurred to me but I skipped it as it seems more complex than
other solutions.

You won't hurt anything with the resistor, just don't go so low in value
that you draw more than the transformer is rated to produce, I would try
something around 470 ohms 1/2W, that's a common value. That'll draw about
25mA and should silence the beeper immediately.