Good as Gold can't normally be copied by Doodle.
To test this, I had a Ditto transform into my Grafaiai and use Doodle to attempt copying Gholdengo's ability. Good as Gold prevented it.
However, Good as Gold can be copied by Doodle with other methods that suppress or ignore abilities.
Mold Breaker/Teravolt/Turboblaze/Mycelium Might
I used a Ditto again for this. Ditto transformed into Grafaiai, used Doodle to get the abilities of Axew/Zekrom/Toedscruel respectively (I didn't test Reshiram but it should still work), and switched into Gholdengo. Doodle was able to copy Good as Gold.
Neutralizing Gas
Since Doodle can't copy Neutralizing Gas, I had to do a local Double Battle for this. Under the effects of Neutralizing Gas from Koffing, Grafaiai was able to obtain Good as Gold with Doodle. Status moves worked under Neutralizing Gas, but once Koffing switched out, status moves were blocked by Good as Gold.
Gastro Acid
Good as Gold normally isn't affected by Gastro Acid. However, in another Double Battle with Neutralizing Gas present, both the opposing Ditto (transformed into Gholdengo) and Grafaiai obtained Good as Gold via Doodle. Still under Neutralizing Gas, Gastro Acid hit the opposing Gholdengo (tera normal). When Koffing switched out, Doodle/status moves hit the opposing Gholdengo, but not the opposing Grafaiai. Both of them still had Good as Gold.
It's a complicated process, but basically, Gholdengo was hit with Gastro Acid while Neutralizing Gas was up. When the Gas went away, Gholdengo was still affected by Gastro Acid, allowing Doodle to copy it. It's a bit redundant, since if you already have Neutralizing Gas, you can just Doodle Good as Gold from the start, but nobody's realistically gonna be using Doodle in the first place so I digress...
TLDR: Doodle can copy Good as Gold only when Mold Breaker/Teravolt/Turboblaze/Neutralizing Gas is present, the user of Doodle has Mycelium Might, or the Good as Gold Pokemon is under the effects of Gastro Acid.
