Using some examples that I found online, I wrote the following VBA that accomplishes what I need. I placed an empty pivot table on my summary page, and a button linked to this macro that updates all other pivot tables with the same filters.
DISCLAIMER: This will not work for fields that have "(All)" Selected. It wasn't necessary for me to look into, but it's possible to modify the following code to include "(All)" support.
Sub UpdateAllPivotTables()
Dim ws As Worksheet
Dim wsMain As Worksheet
Dim ptMain As PivotTable
Dim pt As PivotTable
Dim pfMain As PivotField
Dim piMain As PivotItem
Dim pi As PivotItem
Dim pf As PivotField
Dim hadError As Boolean
On Error Resume Next
Set wsMain = ActiveSheet
Set ptMain = ActiveSheet.PivotTables(1)
Application.EnableEvents = False
For Each ws In ThisWorkbook.Worksheets
If (Not (ws.Name = wsMain.Name)) Then
ws.Unprotect
For Each pt In ws.PivotTables
pt.ManualUpdate = True
For Each pf In pt.PageFields
For Each pi In pf.PivotItems
If (Not (pi.Visible = ptMain.PageFields(pf.Name).PivotItems(pi.Name).Visible)) Then
pi.Visible = ptMain.PageFields(pf.Name).PivotItems(pi.Name).Visible
End If
Next pi
Next pf
pt.ManualUpdate = False
Next pt
'ws.Protect
End If
Next ws
Application.EnableEvents = True
End Sub