and install full secondary application servers in each site.
Deploying the Primary server into the corporate data center, and leaving it clientless, is a good approach.
Then the next question from this then becomes: Where will you have administrators who use the Patch Manager console? In all sites, or only in the corporate site?
If you have a distributed patch administration environment, additional application servers can be useful; if you have a centralized patch administration environment, then likely the only question of concern is whether you connect consoles directly to the PAS in the data center, or deploy a second application server at the central site and connect consoles locally.
Generally speaking, additional application servers are deployed to support additional console sessions, to avoid launching a console session across a WAN-based connection, or where additional security boundaries are needed.
Additional management servers are deployed to support distributed data storage of inventory data, or where unique management scopes (e.g. AD Domains or WSUS environments exist). I suspect in a Configuration Manager-based six-site environment, additional management servers will not serve any useful purpose. As noted, if you are managing Software Updates from each side independently, there may be value in having an on-site application role server.
You may find some benefit from these supplemental materials posted here in Thwack:
Patch Manager Architecture – Deploying Application & Management Role Servers
Patch Manager Architecture - Deploying Automation Role Servers
and possibly for system inventory. The second would be less important since we have SCCM in our environment and it seems like much of that data would be redundant.
Correct, the software/hardware inventory available from Patch Manager is mostly redundant with what you can get from Configuration Manager.
However, a significant difference would be that the inventory selection (i.e. what you inventory) is highly customizable with Patch Manager,
and the reporting engine is significantly more user friendly than is the ConfigMgr reporting
This may well be sufficient justification to utilize Patch Manager for asset inventory rather than Configuration Manager.
However, it is the specific question of performing WMI-based Managed Computer Inventories, and/or other client-management tasks,
that will directly drive the need, or lack thereof, to deploy Automation Role servers in the remote sites.
The good news is that you can ADD those site-based Automation Role servers at any time.
So, it's perfectly conceivable that you might defer that decision until such point that you've identified an express need for the capability.