(famous last words, "what does this post
have to do with electronics repair"...)

A lot... if the circuit damages something.

About 50 years ago, MAD ran a parody titled "Beatnik Magazine". One of its
contents was a photo of a middle-class family, watching TV in the living
room, with the caption "What's wrong with this picture?" The answer was
"Like, man, /everything/ is wrong with this picture."

That pretty applies to the circuit. I see what you're trying to do, but
you're going to get two exploded caps (at least), and possibly a small fire.

Take the other poster's advice. Buy a second detector and plug a 120V alarm
of some sort where the lamp goes. If the one you have an handle enough
current, you could always stick a plug adapter in the lamp socket.