When the migration activity runs(weekly) from past 2 times , we saw the cluster read replica instance has restarted as it fallen behind the master(writer instance).
I can't figure out what your setup is here. You must be using logical replication (not physical) or you wouldn't be able to write to the replica at all. But if you are using logical replication, why do you also need these weekly jobs? Why isn't logical replication taking care of it?
Everything
after that worked seamlessly but we want to avoid the replica getting restarted. To avoid from restart we started doing smaller binary files and copy those files to the cluster-2
Who restarted it? I am not aware of any case where the replica responds to falling behind by restarting itself. With physical replication, it can start cancelling queries, but you don't seem to be using physical replication.