Hi AMM,
Associated Views cannot be used as the default view of a sub-grid like public views, which is by design.
"We want to break out some of the related entities onto a different tab but with that same view and features such as the filters."
- It's impossible to implement exactly the same Associated Views in a new tab, but we can create a sub-grid in a new tab to mimic Associated Views as much as possible.
Create a new tab, and create a new sub-grid in it. In the Data Source, select Records as Only Related Records.

Let's look at the difference between them.
This is the new tab with a sub-grid.

This is the original Associated Views.

They show the same records and have the same command bar buttons.
The differences are 1) Associated Views has "Due:" to quickly filter by time, 2) switch whether to include related, 3) search records, 4) switch views quickly
So if you have no requirement for these four features, then you could insert a sub-grid in the new tab to achieve your needs.