Currently, it does not appear possible to change a GridSplitter's Template via a Trigger.
This manifests as the "Apply Resource" option being greyed out, when the GridSplitter does have one, or many Templates available.

See attached screenshot.

Repro Steps:

1) Create a Studio Project featuring a MainPage.xaml

2) introduce a GridSplitter Element to the Root Grid, and rename it to give it a custom name.

3) Right-click on the GridSplitter, and create a couple visually distinct Templates for it, saved to GlobalResources.xaml

4) With the GridSplitter selected, open the Interactivity/Triggers Panel, and create a new DataTrigger on the GridSplitter.

5) In the Trigger's 'Binding' field, click its 3-Dot menu, and select 'Create Binding'.

6) Select the the 'Relative' Tab, select the 'Name' tab, and in the 'Element Name' field, select the name of the GridSplitter given in Step 3).

7) Add an 'Action' to the Trigger, setting the 'Target' to the name given to the GridSplitter in Step 2), then set a 'Property Name' of 'Template'.

8) Still within the Trigger's Properties Panel, hover over the 'Value' field, and click on its 3-Dot menu to open it.

Observe how the "Apply Resource" button is greyed out, and cannot be selected, despite two valid 'Template' Resources being available to the Element.

Expected Results: GridSplitter Templates are available for application from the Trigger Action's 'Value' field.