I’d be more inclined to use regset or regset64 because if you want to have the real value in there and not the environment variable then you can store it as a REG_SZ - no need for REG_EXPAND_SZ.
However, you are putting the environment variable in quotes - you can see them in your regedit screenshot. Take the unnecessary quotes away and it will work.
Why do you want REG_EXPAND_SZ to store a literal? REG_EXPAND_SZ is used to expand an environment variable to its value - not needed here because you said you wanted to store the literal computer name.
If you insist on it being REG_EXPAND_SZ, and the expansion, you need to obey all the rules and append a null character to the stored string. To do this you will need to use the reg command directly or via a .reg file that you create.
Export a known good version to a .reg file.
Trim out any values you don’t want o affect.
Upload the .reg file.
Create a fixlet or task that downloads the .reg file and use reg.exe to import it.