Hello Lisa,
The App Engine Deployer role is the recommended role to grant to the account responsible for deploying a new version of a service, although you could also deploy with the App Engine Admin role. Note that other roles may be required to deploy, depending on which configuration files you are deploying; this is detailed in the deployment instructions below.
In the same article,
some checking steps are recommended to troubleshoot IAM-related deployment failures. A list of App Engine roles and their descriptions are listed in
this article.
After reviewing the articles, if you still cannot deploy your app without using the Owner role please send me your project number (privately) and the error messages that you receive and I'll glad to investigate the issue further.
Sincerely,