I have to run a scheduled batch class in a sandbox to update records in production. Please let me know if that is possible some way.
I can't create the batch class in production due to some business restrictions.
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Well that certainly is a complicated situation but sticking to your question, I think your best bet would be to make the batch use the SF API, just as @kurunve said.
Make the batch with the logic you need and then, wherever you're supposed to do a
Database.update(myrecords), then make a callout to Production enviroment.
Here you can find extense documentation for the REST API.
Here you can find a trailhead that guides you through the process of making such a callout.
Take into account that you will have to register the endpoint which you will callout to in
Setup > Remote Site Settings.