how set custom search to also include non-public google site?

Question

We have a non-public Google Site for our local organization volunteers who must be signed into our google workspace for access. The site is non-public to protect some organization confidential materials. We also have several external sites indexed by Google.
The non-public Google Site search is via the default search [magnifier icon] upper right which returns only the private site content.
Volunteers need an integrated search of both the private site; and several of our public sites.
We have added a programmable search widget on the non-public site that restricts external search to the designated external sites and it works pretty good; but does not include results from the private site itself.
So volunteers have 2 different search widgets: site search via magnifier vs external custom search on our external sites
Question: how can we setup a single programmable search widget which returns combined results from both the non-public site and the several external sites.  NOTE: the user is always authenticated / log'd in to the non-public site when using the programmable search widget
non-public site is https://sites.google.com/scorevolunteer.org/cleveland-volunteer-portal/search-score [sorry, restricted only to logged in users of scorevolunteer.org]
programmable search widget on the above site limits Google external search to score.org/cleveland and to score.org