Concerning your question:
Can my theme be submitted for approval with these errors?
I’m assuming you’re asking whether your plugin, while throwing an error upon activation using TGM library, can still be submitted and accepted to official WordPress repository. My answer would be, no, it cannot.
It seems like you made your plugin work once it has been previously installed, but activating it in its current state throws errors. People reviewing plugins will reject it, if they encounter errors upon just the installation and activation, and if you do it a couple of times in a row, your plugin might be permanently rejected from the repository. Therefore: test, test, test. For more information, please refer to WordPress approval workflow
If you’re new to creating WordPress plugins, I would strongly advise that you follow some boilerplate, that will show you the best practices and allow you to create a quality plugin, with all the features like localization and multisite-support. My recommendation would be WordPress Plugin Boilerplate