Well, the British did accept mediation over a serious dispute with the Empire in 1862. They did not respect Beligum's favoring of Brazil, but they still gave it a try. If you kill or temporarily disabilitate Mr. Rhodes we could see it.The problem is how one gets the British to accept arbitration. Before and after the Ultimatum the Portuguese tried to convince the British to accept a neutral mediator. They weren't willing to compromise and one can understand why: they wanted it and they took it.
If only the territory was of little value or there was any level of uncertainty of the outcome of a war between the parties in conflict, the British would have accepted mediation...