I have a custom helm, with a custom, blank enchantment attached to it.
In my script I am trying to add an effect to the enchantment, but nothing seems to work.
The script compiles, but when I reload the esp in the CS, it reports an error saying that the reference (in the AddFullEffectItem statement) is invalid.

The code I am trying is:
AddFullEffectItem SHLD 5 0 0 0 helm

I also tried:
set enchant to GetEnchantment helm
AddFullEffectItem SHLD 5 0 0 0 enchant

When I run the mod despite the error that is reported, the script crashes.

I also tried to test from the console in game, but it doesn't seem to recognize the effect codes.

I can't seem to figure out how to use this function.

Use AddFullEffectItemC. The 4-letter codes for magic effects are actually editor IDs. Annoyingly, the game assigns a formID of zero to magic effects unless they are edited by a loaded plugin, preventing them from being used reliably in scripts. This is why OBSE provides 'C' versions of all its magic effect-related commands, and why you should prefer to use them instead.
More annoyingly, we don't currently provide an easy way to get an effect code for a magic effect with a formID of zero. Next release of 0019 will include MGEFFromChars to rectify that.

I see, thank you.

So the following should work? ...

let code := GetMagicEffectCode SHLD
AddFullEffectItemC code 5 0 0 0 helm

No, that's the thing about magic effects: when you pass in SHLD as an argument OBSE sees it as the number '0' representing its formID, so we have no way of identifying the effect setting.
0019 beta 4 was released about 15 minutes ago. If you are interested in testing with it you can do:

let code := MagicEffectCodeFromChars "SHLD"AddFullEffectItemC code 5 0 0 0 helm