Your Camera as you say is only able to run about 3.6w and that siren is 15w or same as 1.25amps. so that would require you to use your output to work with another relay that has more then say 120 Ohms relay coil.. If it is 40 Ohms it might be able to drive it but could be not reliable so you would want to keep it at or over 120 Ohms.. Then you would want to use a 12v power supply with your external relay.. That power supply you would want to be no less then 1.5amps as you never want to run a power supply at the max output. However because you won't be driving the siren for extended time and only short times a quality 1.25amp power supply normally has a higher ability and could be fine but again if you are buying new then just buy a 1.5amp or better 12v power supply..
Edit, If it was a Dry contact and not a wet contact you would be able to get away with just a 12v power supply but seeing it is wet you will want an external relay or as stated before a powered siren that can work with your Wet Contact setup..
Ok to be honest, There are a few different things that might be going on. With that one I looked at the full listing and it does say will work with 320ma. Where yours is only 300ma not sure it would work.. However in case that kind tells me a few different things. One that the system only requires 320ma to make the siren sound but again with doing the math 300ma is only 3.6w at 12v so could mean they are calling it a 15w Max but can be driven at a lower volume starting at 320ma in their case and that would be 3.84W and could be a Piezo and not a Coil siren and is it was then it might be same or greater then 15w coil siren..
4pin 12v Bosch style relay.. The issue with a POE splitter is that it takes the POE and gives you only Ethernet and 12v so you would then need a 12v Y as well and power the camera on 12v as well if you went that route.
Could use a 5pin and just cap off 87a Either way did a rough draft of a relay and not all relays have Diodes so you would want to make sure that you put one in between 85 and 86 because when the relay switches off you need it to clamp any voltage that could damage your camera..