Async replication via BackgroundService Extension

### Is there an existing issue for this?

- [x] I have searched the existing issues

### Is your feature request related to a problem? Please describe the problem.

Split out from #1181, see discussion in #1182. Related to @NehanPathan's GSoC 2025 work.

It was determined to be infeasible to add async support to our existing ReplicationClient, so we should create a new approach.

### Describe the solution you'd like

We should add to the new [lucenenet-extensions repo](https://github.com/apache/lucenenet-extensions) support for an asynchronous replication client. This should be implemented as a [BackgroundService](https://learn.microsoft.com/en-us/aspnet/core/fundamentals/host/hosted-services?view=aspnetcore-10.0&tabs=visual-studio) so that we don't have to reinvent the plumbing (like was done originally in ReplicationClient). BackgroundService will also automatically let us use Microsoft.Extensions.DependencyInjection for this.

### Additional context

_No response_

Start by reading the discussion in #1181 and #1182, then review the new lucenenet-extensions repository and the linked Microsoft BackgroundService guidance. Define the asynchronous replication client as a BackgroundService with Microsoft.Extensions.DependencyInjection support; done means the extension provides this new replication approach.

Written by the indexing model from the issue text.