Currently Bamboo doesn't have the option to add the submodule ssh key when saving the main repository.
These keys have to be manually added inside each submodule repository on Bitbucket.

It would be practical whenever a customer enables “Use submodules” for a linked repository, Bamboo should run a Git command to list all of the repositories that are added as submodules inside the .gitmodules file, and, for each one of those repositories, add the SSH Key for access.

or 

Have an option in the Linked Repository to add the submodule repository.

Similar request as https://jira.atlassian.com/browse/BAM-25617