Paint scaffold persistent bottom sheet background color behind the keyboard - #186365

Fixes the framework-owned part of #179482.

Problem

With a persistent bottom sheet (Scaffold.bottomSheet or ScaffoldState.showBottomSheet) and resizeToAvoidBottomInset: true, the sheet is lifted above the keyboard and the Scaffold's own Material (Scaffold.backgroundColor) is what sits behind the keyboard. On iOS 26 the keyboard is translucent, so the scaffold color shows through under the sheet instead of the sheet's color.

Fix

Lay the sheet out all the way to the bottom and pad its content, instead of lying about its size:

• _ScaffoldLayout extends the persistent sheet's slot by the part of the keyboard inset that is not already covered by the bottom bars (bottomSheetKeyboardInset), and keeps reporting the content-only height so FAB and SnackBar geometry are unchanged. The inset reaches the pre-built sheet widgets through _BottomSheetBoxConstraints + a LayoutBuilder, the same pattern as _BodyBoxConstraints / _BodyBuilder.
• New BottomSheet.bottomInset (default 0): a Padding inside the sheet's Material, so the surface (color, surface tint, elevation, shape) continues behind the keyboard while _childHeight, the drag/fling dismissal thresholds and constraints are still measured against the content. The Padding is always present so the content's element tree and focus survive the inset changing.

Because the sheet is laid out within its slot, a Scaffold shorter than the keyboard keeps the sheet inside its bounds without any clipping.

Behavior notes

• Only the sheet's own Material color extends (showBottomSheet(backgroundColor:), BottomSheetThemeData.backgroundColor, or the default surface color). A child's Container(color:) does not.
• With the keyboard up, the sheet slides behind the keyboard glass on open/close rather than emerging from above it.
• The extended Material absorbs taps below the content, which is what the Scaffold's Material did there before. Only a docked keyboard produces an inset.
• The sheet's semantics container rect now includes the extension; child rects are unchanged.
• resizeToAvoidBottomInset: false, showModalBottomSheet, and a BottomSheet passed directly as Scaffold.bottomSheet behave as before.

Tests

No new test file. persistent_bottom_sheet_test.dart gains a keyboard inset group (11 tests): extension with and without a keyboard, resizeToAvoidBottomInset: false, bottom navigation bar, docked FAB, focus preserved while the keyboard appears, fling and slow-drag dismissal thresholds, constraints, taps in the extension, a Scaffold shorter than the keyboard, and a nested BottomSheet not being extended twice.

bottom_sheet_test.dart gains a BottomSheet.bottomInset group (5 tests) that exercises the new API on a standalone BottomSheet: layout (plus the default value and the non-negative assert), constraints (min and max) bounding the content rather than the inset, the drag-to-dismiss threshold measured against the content through a real AnimationController, state and focus preserved across inset changes, and showModalBottomSheet not setting it.

Pre-launch Checklist

• I read the Contributor Guide and followed the process outlined there for submitting PRs.
• I read the Tree Hygiene wiki page, which explains my responsibilities.
• I read and followed the Flutter Style Guide, including Features we expect every widget to implement.
• I signed the CLA.
• I listed at least one issue that this PR fixes in the description above.
• I updated/added relevant documentation (doc comments with ///).
• I added new tests to check the change I am making, or this PR is test-exempt.
• I followed the breaking change policy and added Data Driven Fixes where supported.
• All existing and new tests are passing.