Veeam Backup and Replication solved all my problems.

Veeam is incredible at handling full backups without impacting my network. Large full backups can be done using synthetic backups from incremental. We are a small to midsize company and cost is always a factor. The product is reasonably priced, but I need to make decisions regarding some less critical machines to not back them up to keep the license count lower. If there was a limited license option that could be scheduled out for a once-a-month style limited backup, that would be ideal. It is a very minor complaint though, as the licenses aren't really that expensive.