It is DC. When using a meter it wil Detect AC over DC since the current is mostly AC. The wide chip at the bottom of the relay is selenium a rectifier.
If you remove the transformer cover you will find more of the round chips or diodes.
The theory is the pulse activates the relay and that activates the whistle. Most buttons have two pulses. The first is high to start the relay and motor, then the second just continues the sound.
Wow, You need DC so your options are limited. Fix it, get a sound button or another transformer. Considering what the whistle buttons go for you might be better off with an extra transformer. A key part is heat senstive, relay?in a bulb. It may not be available.