If you name a light_spot, it automatically becomes a dynamic light and you can do fun things with it. As far as I know, parenting it to a moving entity so it will move is one of them. There's a hack to get continuously rotating entities into TF2 (since func_rotating was hacked out of it to begin with) but I don't remember what it is off the top of my head.
You can't really do that with light_spot. Source does dynamic lightmapping by baking in every combination of lights being on/off, and switching between them when the lights turn on/off.
The entity you probably want is point_spotlight. Then you'll want to make it rotate. Func_rotating is broken in TF2, so my personal recommendation is to use a func_rotating_door and set its distance to 999999999999 (or some other absurd value) so that it theoretically won't stop spinning for at least a month or so.
However, I've had issues in the past with parenting point_spotlight. If that method looks wrong, then you'll probably want to find an existing Valve map, find a decompiled version of it, and look at how they did it. My guess is that it's a particle effect.
There's actually a particle specifically for this prop:
So just place an info_particle_system entity in your level,
make the ParticleSystemName keyvalue: mvm_emergencylight_glow
You can give it a StartActive keyvalue to make it auto-play on map start, or give it the Start input from some other output source, and bam, done.
is the properties of the beacon
look at the model's animations and set the prop_dynamic's default animation to the relevant one.
I think the one for that emergency light thingy is "spin"