Refactor `MultichainRouter` to not use `withKeyring`

`withKeyring` acquires a mutex lock when sending a request, that means that anything that requires the `withKeyring` in one of the Snap RPC calls (like `snap_getBip32Entropy`) will be hanging due to the deadlock that the router creates.

As of today, a temporary fix has been done on the extension side to avoid that issue: https://github.com/MetaMask/metamask-extension/pull/32120

Start by tracing MultichainRouter's use of withKeyring and the mutex acquired during request sending. Reproduce or inspect the deadlock involving the snap_getBip32Entropy RPC call, using the linked extension pull request for context. Done means the router no longer deadlocks when a Snap RPC call requires withKeyring.

Written by the indexing model from the issue text.