I agree with the comment by @notovny. Space isn't (necessarily...) cold. In a certain sense it is hard to talk about temperature of space since it isn't a material. What we usually mean when we talk about temperature in space would be the temperature of any ambient radiation. Some things that are true are:
As @notovny pointed out, space is insulating. So if you want to keep something cool in space you need to be able to radiate thermal energy away. This only works if the object is above the ambient radiation temperature.
If something in space is shaded from sun (like the working bits of James Webb Space Telescope, or any low Earth orbit satellite on the night side of Earth) then they will behave roughly like the ambient radiation temperature is very cold. More or less we are talking about the 2.7 degree temperature of the cosmic microwave background (near Earth there would also be "Earth shine" which would increase this). This is probably what gives the impression that "space is cold".
If something isn't shaded then it will receive energy rapidly from the Sun (2000 J/m^2 roughly if in Earth orbit). This could be radiated away if the radiators are shaded. But now you have some difficulties with the relative size of the radiators (if too big then hard to keep them shaded, if too small then radiation rate will be smaller than heat received from the Sun...).
An orbital gun like you are talking about will have to generate and use huge amounts of electric power. However you generate it, and however you convert it to kinetic energy of the payload you are accelerating, there is going to be a lot of waste heat. All of that has to be radiated away. If it is in Earth orbit then radiating that away will be particularly difficult any time it is on the day side.
So I think the main benefit you see, ("space is cold") may not be much of a real benefit. The difficulty you identify (catching the payload at the other end of the trajectory) would be no different from if it was accelerated using chemical rockets, or any other means.
The other problem you are neglecting is recoil. When the gun accelerates the payload the gun will be slowed down in its own orbit (assuming it is accelerating the thing "forward" in the orbit). So during and after each shot it would need to do something, like firing rockets, to keep its orbit from decaying with each shot. Given that it is likely more massive than the payloads it is shooting, there might be little saving in rocket fuel. Time to do some delta v calculations...