Me and my friend tried this but he still can't ready up. Same issue if he hosts, I get a message saying I don't have the map and it's workshop ID number. This applies to all the workshop maps we've tried

The map itself is made for multiplayer. I think the issue you encounter is a bug in the workshop support of the game. You can copy the map and all other maps you want to play in MP to the regular old MP map folder to fix this.
The workshop maps for RA3 are in here: C:\Program Files\Steam\steamapps\workshop\content\17480\ (assuming you installed Steam in C:\Program Files). Sub folders there are the workshop items. The Brainfreeze map is 3446209067.
The regular old map folder is here: C:\Users\<your windows user name>\AppData\Roaming\Red Alert 3\Maps
Copy the Brainfreeze map folder from workshop location to the 2nd path and it should be a normal MP map. The bugs the game has with it will switch from "You dont have the map" to not displaying its name in your map list :-)
We told the developers about all the new issues there are since workshop support was added, but I dont know if they are working on it.

I can't get this map to function with my friend online. I can play solo but my friend can't ready up for me to start the map when playing online in a lobby. We're both subscribed to the map too

Thank you very much! It works

There is no button in the client sady. I used this guide to find out how to use a command line version of Steam and used it to update the map: https://psteamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=3435449860

Hello, dear author. Tell me please, how did you updated your map in Steam Workshop? Which button in steam client?