To manage file type filtering for Teams emails and files stored on Teams' shared storage, you might consider upgrading from Microsoft Teams Essentials to a higher-tier Microsoft 365 or Office 365 subscription that includes Exchange Online and SharePoint Online. This will allow you to leverage the full capabilities of Teams, including file management and email integration.
Since you are currently using Exchange on-premises, transitioning to Exchange Online would also be beneficial for managing client emails in the cloud. Licenses such as Microsoft 365 Business Standard or Microsoft 365 Business Premium could be suitable options as they include Exchange Online and SharePoint Online, which are necessary for the features you are looking for.
Make sure to evaluate the specific features of each license to find the best fit for your organization's needs.