[Due for payment 2025-08-20] Use react-native-reanimated in POPOVER Modals

[mountiny]

Part off #64775

Update the POPOVER modal type to use react-native-reanimated under the hood.

Issue Owner

Current Issue Owner: @clarajones-expensify

[melvin-bot]

Triggered auto assignment to @clarajones-expensify (Bug), see https://stackoverflow.com/c/expensify/questions/14418 for more details. Please add this bug to a GH project, as outlined in the SO.

[jmusial]

Tracking of types / PRS comment for POPOVER modals:

• - SearchRouterModal - Migrate Popover modals to use react-native-reanimated #67748
• - Popover - Migrate Popover modals to use react-native-reanimated #67748

[mountiny]

WIP