@docsearch/react: Remove dependency on @base-ui/react

## Describe the problem

`@base-ui/react` adds too much weight. It was short sighted to add just for 2 components, even if it does make a11y easier.

## Describe the solution

Will have to explore either smaller alternatives, or just manage it all in package.

## Alternatives concidered

We could lazy load the `FacetBar` and `SourcesPanel` components, but that only delays the bundle size being loaded and introduces potential lag on a user's interaction.

Start by locating the @base-ui/react dependency and the FacetBar and SourcesPanel components. Compare smaller alternatives or an in-package implementation while preserving accessibility and avoiding the deferred bundle cost; done means @base-ui/react is removed without interaction regressions.

Written by the indexing model from the issue text.