New Case Contact Table: implement "filter" button/functionality

[MikeRose151]

What type(s) of user does this feature affect?

• volunteers

Description

This ticket is a sub-section of a bigger ticket: #6462.
Some foundational work for the bigger ticket has been started in this PR: #6467.

This ticket is to add the "filter" button and its functionality to actually filter the table by its constraints. The existing version of the case contacts table (old design) includes the logic for filtering, so it should be a case of refactoring so that it works on this new design.

Screenshots of current behavior, if any

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