Thank you, that did somewhat solve my problem. I was able to sync data with a specific condition to a separate base now.
I should have mentioned I am trying to be able to do this for various different conditions on the same view. That solution does not fully accomplish this because I would have to manually adjust filters and re sync.
For example, I want to have a main base with multiple views that has data for various clients pertaining to the view. This would be synced to an admin site, where they can sign in and manage their data. But then when they edit data, I would like it to sync their data to their specific base, which will then be synced to their personal site.
After looking further into it (Sorry, very new to Airtable), I realized that within a table I can have the main combined view and then also have a separate view within that table for each specific client, filter those appropriately, and then sync those client views to their appropriate base.