How To Post 100% discount condition type value to a different profit center in the same GL account

Hello Experts,

We have a requirement from the client where they are offering free of month offer to the serives for they are providing.Document flow is from contract->Invoice.They have periodic billing in the contracts and this offer will be only given for the 1st month of the billing plan.Now we we are following two approaches

1. Client wants to go with creation of credit memos only for the 1st billing month and credit the amount back,and from the 2nd month normal billing will happen.After accounting document will be created the want to post that free of month amount to a different profit center.

Question

Can this be achived using the above process.What can be the logic to credit trigger credit memo only for 1st billing month and how the amount can be posted to a different profit center for the same GL account.

Also we are trying for another approach.

2. We thought of creating a 100% discount condition type which will be applied at the invoice level and will be triggered for only the 1st billing month.But the issue we are facing is being net value zero for the billing document no accounting document will be generated, so how the discounted value to be posted in the same GL acount but different profit center.

And also is there a possibility of creating a Accounting document with zero net value.?

Please suggest if there is any config done for it and any other way to achieve this functionality.

Thanks.