Streaming Sporting Events

I am an audio noob, and am looking for some advice.

We have started streaming swim meets and are rapidly getting overcome by audio equipment. Our pool is set up with it's own built in announcement system that uses a mic with an xlr cable. We are trying to split this mic so that we can still make announcements in the pool but also have it go into our computer for streaming. We would like the ability to have the one mic feed either or both systems. We have the ability to mute the stream side within the software, but there is no control on the pool side.

We have tried a simple splitter cable and rapidly learned a bit about phantom power, plus we didn't have the ability to mute either output. We're on a budget, so want to make sure we get what we need this time. Do we need an audio interface or a sound mixer, or can we get by with something else? Any recommendations on specific gear?

Thank you for your help!