How to get a registration token for Firebase Cloud Messaging without requesting permission for notifications?

There is currently no other alternative to get a registration token nor is there a different way to subscribe to topics for a web client.

You already know this, but might as well mention it. Requesting permissions is a must for security purposes (preventing notifications not wanted by the user) and that in order to subscribe a web client to a topic, you'll first have to get a token (which won't generate unless the user grants permission).