[Due for payment 2026-05-12][Search Filter Revamp] Release #1 Update the filter bar to use the searchbar position & add the buttons for filtering/saving the search.

[JS00001]

Coming from #81558 (and the mockup here: https://exfy-newdot-filters-exclusive-v3.netlify.app/reports and here)

Lets start this off by updating the current filters bar to look like the mockups:

Note, this will not include any changes to the filter chips (including the x icon/split filters). This will only move the searchbar & move the button for filters, display, & save search. The 'x' in red should not be added.

Issue Owner

Current Issue Owner: @bernhardoj

[melvin-bot]

Current assignee @trjExpensify is eligible for the NewFeature assigner, not assigning anyone new.

[melvin-bot]

Triggered auto assignment to Design team member for new feature review - @dannymcclain (NewFeature)