Since your Org-Wide Default for Account is now Private, you'll need to use Sharing Rules and the role hierarchy to control the visibility of specific subsets of your Accounts.
Make sure to turn off View All on the Tester profile, because that will enable the testers to see all Accounts regardless of sharing. Also be aware that Record Type enablement on profiles only controls which record types the user can create; it has no effect on record visibility.
Record-level sharing does not operate on Profiles: it uses Roles, Public Groups, and Users as the recipients of visibility grants. You'll need to establish one of these groupings to hold your Testers; it could be either a Role or a Public Group depending on your organization's existing role hierarchy. Then, you're going to need a Criteria-Based Sharing Rule to share Accounts to that Role or Public Group based upon their record type.
Provided your Testers aren't also at the top of your role hierarchy (you'll want them at the bottom of the hierarchy so they don't inherit visibility), this should get you the visibility you need.
This is actually just vanilla Salesforce functionality. The NPSP doesn't alter it. However, the NPSP's householding function assigns all Contacts to Household accounts by default, so if your Testers have the right to see Contacts (and they're not set to Controlled by Parent), the Tester is going to get a parent share to the Household anyway. You may need to further adjust your settings for Contacts.