I'm answering my question because I found the solution for this issue. I hope this helps anyone else who runs into this, because at the time there was little info on the web describing to how to solve this.
It turns out that my user account did not have "owner" access to the bucket. Here are the steps I took to grant myself access:
1) First, navigate to your project's Cloud Storage Browser in the Developer Console.
2) Once you see a listing of the buckets that are linked to your project, check the box next to the bucket(s) you'd like to modify the permissions for and then click the "Bucket Permissions" button.
3) Next, add your user account to the list of permitted users. Set the permission level to "owner". Click the "Save" button when you're done.
You should have access to the bucket now, which means you won't run into any 403 errors. If you are, you did not set the entity correctly or are using a different account when you authenticated with gsutil. Double-check your work and try again.