This should be pretty easy then
Bring up the 2016/2019 server.
Ensure you have a trusted 3rd party certificate on the new server:
https://learn.microsoft.com/en-us/exchange/certificate-requirements
Ensure you have the necessary FW ports open between the new and Exchange online ( Mimic the existing setup)
Run the Hybrid Wizard and point to the new server as the endpoints.
Create a new new custom receive connectors on the new server if they exist on the old.
Ensure the new server is set as a source server in all the send connectors. Remove the old server and verify mail flow.
Move the arbitration mailboxes from the 2010 server to 2016
https://www.alitajran.com/move-arbitration-mailboxes-in-exchange-server/
If all looks good, uninstall the 2010 server.