Probate is the process by which the state administers the estate to make sure ( i.e." to prove") that all debts are settled before the estate passes to the beneficiaries. The probate process includes a determination of whether or not there was a will, whether any wills or codicils are valid, and the appointment of a personal representative (executor) to gather the decedent's property that requires probate, pay creditors and expenses, and distribute the rest to the rightful persons (beneficiaries).

The translation of "executor" as "Testamentsvollstrecker" is problematic because their tasks and functions differ. I personally prefer "Nachlassverwalter".

That said, I would translate your sentence as follows:

"Ich vermache XXX den Betrag von 2000 Pfund, und zwar unabhängig davon, ob er in seiner Eigenschaft als Nachlassverwalter gerichtlich bestätigt wird oder nicht. "

( in anderen Worten: diese Summe steht ihm zu, egal ob er Nachlassverwalter ist oder nicht)