I fear the production DB needs more transaction log space for those batch DELETE's than it has at its disposal.
Let me give an example:
In one of my projects I have built a process in PowerCenter to export huge numbers of workflows from huge folders and to import them into another environment.
The export was no problem at all.
But while importing those huge XML files we received error messages from the repository DB that there was not enough transaction log space available to keep all those records for this huge transaction "alive".
We had to ask the DBAs to change the transaction log size from 2 GB to 10 GB for this process. Then everything worked fine.
Please ask your DBAs to check whether the problem is indeed the transaction log size (which can be set for each database on a SQL Server machine individually, but of course it has to fit into the existing disk space).
If so, the remedy is to increase it. How much? I can't tell, that depends on the size of the records to be deleted.
If that doesn't work for whatever reason, there's an alternative, but I would prefer to discuss this only if needed. No use in discussing hypothetical processes by now, I think.