refactor: roll out new RPC client in `BitcoinRegtestController`

[fdefelici]

In #6250, a new Bitcoin RPC client was introduced.

This issue tracks the refactoring of BitcoinRegtestController to replace the existing BitcoinRPCRequest with the new RPC client.

Since RPC calls are widely used, the migration will be done incrementally across multiple PRs to keep the review process manageable.

This issue will serve as a reference for those PRs and will be closed once the roll-out is fully completed.