Start a firestore transaction without a transaction.get()

Well if you do not want to read the document, you should be using Batched Writes.

Firestore Docs:https://firebase.google.com/docs/firestore/manage-data/transactions

Batched writes

If you do not need to read any documents in your operation set, you can execute multiple write operations as a single batch that contains any combination of set(), update(), or delete() operations. A batch of writes completes atomically and can write to multiple documents.