Some quick googling leaves me without any answer to this. (Rather, the results are seven years old and based on the old MC launcher)
Using the new Minecraft launcher, I made a new instance for Forge+mods, ran it, loaded up a world etc., then tried pointing the Forge installer at the new folder.
The installer.log is four lines, so I just pasted it here:
JVM info: Oracle Corporation - 1.8.0_241 - 25.241-b07
java.net.preferIPv4Stack=true
Found java version 1.8.0_241
There is no minecraft launcher profile at "C:\Users\MyUsername\AppData\Roaming\.minecraft.forge.1.16.1\launcher_profiles.json", you need to run the launcher first!
Creating a symlink to the profile file in the "main" appdata\.minecraft folder seems to allow the installer to do its thing, but it seems like a fairly terrible hack and shouldn't be necessary.
Edit: problems seem to be because you're not "meant" to have a different folder for different instances, no matter how intuitive that seems. After installing in .minecraft things seem to work, whereas before, the installed profile would fail to launch entirely despite the vanilla instance at the alternative location working perfectly normally.
