If you use transactional replication, the impact should be pretty low. Once initialised, the publication does not actually touch the source database (well not as from a record locking perspective). The logreader agent only accesses the database's transaction log. It can be scheduled (just like any other SQL Server Agent job) at any time you wish if you need to make sure that it has minimum impact on prime time processing. Otherwise, set it up to continuously run. If you do schedule it to run off peak, it may impact on the size of the transaction log file - log records must remain in the transaction log until the logreader agent has processed them (into the distribution database).

Whilst I am not 100% sure, I think that the issue you are most likely experiencing when the snapshot is being created is blocking - the snapshot process is probably holding a read lock on the table(s). If this is the case, you may be able to reduce the impact by creating several smaller publications - each snapshot would run for a shorted period of time and may block other processes but for a shorted period of time.