You could record it with a clockwork bug
A clockwork bug is a wondrous item which allows sound to be magically recorded onto a gemstone, and then reproduced later:
A clockwork bug can record all nearby sounds within a 20-foot-radius spread onto a low-quality gemstone worth 10 gp, which is embedded in its body. The bug can record up to 1 hour of sound, which must be continuous; if the recording is stopped early, it cannot be restarted. Making the clockwork bug replay the recorded audio is a swift action.
You just need to get the siren to sing while the bug is recording, and then play back the recording later.
This isn't really "stealing" the siren's song since it does not deprive the original siren - it's just a copy, after all - but it is a magically produced copy, and it might count as "capturing" the voice. The text doesn't state anything about the quality of the sound played back later, so it might or might not be good enough to be indistinguishable from the original.
You could have the siren sing into a bottle of messages
A bottle of messages is a wondrous item which allows you to speak a short message into the bottle, which is then sealed with a cork, and then release the message by uncorking or smashing the bottle later:
This green glass bottle has a tiny winding key on its elongated stem that, if turned, causes a shadowy cork to slowly become substantial over a 1-round period. Anything the owner of the bottle speaks into the bottle during this time (up to 25 words) gets trapped inside it once the cork fully manifests. As soon as the cork is removed or the bottle is smashed, this message is released exactly as if the owner had spoken the words at that moment. Once the bottle releases its message, it shatters.
You could give such a bottle to the siren, ask (or demand, or compel) her to sing a verse into it, and then take it away for use in the ritual.
This still isn't "stealing" but seems much more like a direct "capture" of the siren's exact voice, though with the proviso it can only capture up to six seconds or 25 words of a song. It does feature the wording that the message is released "exactly as if the owner had spoken the words at that moment", which you could favourably interpret should be exactly as if the siren was present and singing the song... if more than six seconds/25 words of song are required, you could perhaps very carefully use multiple bottles to capture the song and release them in order later.
A tiefling witch could use the Steal Voice hex
Steal Voice is a major hex available specifically to Tiefling witches, which allows the witch to deprive the target of their voice and additionally to reproduce the voice of the victim:
The witch can steal the voice of one creature within 30 feet, causing it to lose, for a number of rounds equal to the witch’s Intelligence bonus, all abilities that rely on speech, including talking, casting spells with verbal components, and using auditory bardic performances. A successful Will saving throw negates this effect. If this hex is used upon a willing target, the duration lasts for a number of hours equal to the witch’s Intelligence bonus. Additionally, the witch can change her voice to match that of any creature whose voice she has stolen with this hex, as per vocal alteration.
At a glance this seems like exactly what you're looking for, and it even involves actually stealing the siren's voice. It's let down a little by the fact that the "voice-stealing" bit only actually lasts for a few combat rounds in most cases - though that raises to a few hours if you can convince the siren to surrender her voice willingly. The witch technically retains the ability to reproduce the siren's voice at anytime thereafter, though, even once she's recovered from the direct effect of the hex. This doesn't grant the witch the mechanical ability to actually reproduce a siren's song and the supernatural effects that entails, but thematically it's a winner.
If you're one-shotting with premade characters of sufficient level, you might include a tiefling with with this ability and see if the players notice it's there. An NPC involved in the ritual (or available to recruit or hire) with this ability could also work.