So - that site is linked using our legacy setup - not using the GitHub app. This means YOU are already using the deploy key and there is no way to “undo” that as far as I know. So, to fix, I’d suggest you relink the repo to our service in our UI, here:
(choose edit settings, then select “link to a different repository”, but link to the same one).
This will allow us to clone your code without a deploy key (for the main repo), and make using a deploy key for the secondary (submodule) repo easier. Otherwise, thanks to GitHub’s “security” check on reusing keys, you’ll have to instead create a user that has access to all repos and assign the key there, which may not even be possible for you.
Let me know if you can get the repo re-linked and then I’ll generate a new deploy key for use in the submodule only.