If 'totalitarianism' is used in its current sense, so that a totalitarian regime is one that totally controls every aspect of its citizens' lives, there is no significant sense in which Edmund Burke (1729-97) was a totalitarian in his political thinking. There are issues, too, about whether a 20th-century concept such as 'totalitarianism' is anachronistic when applied to an 18th-century thinker; but I set these aside. Historical finesse aside, I understand the question.
Burke as communitarian - not totalitarian
Burke was in modern terms a communitarian, in his case by 'appealing [to] the idea of the national
community ... and [protesting] the lunacy of a world without order, custom, and tradition'. (Douglas Archibald, 'Edmund Burke and the Conservative Imagination', Eighteenth-Century Ireland / Iris an dá chultúr, Vol. 10 (1995), pp. 127-147 : 141.)
There is precious little custom or tradition in a totalitarian regime, which thrives on the radical change or abolition of any institution or practice that does not fit its ideology. For convenience I'll use the formula of 'traditions and institutions' from now on.
Burke as defender of common law rights
Burke believed in the civil or common law rights of English citizens, a tradition which the state should not abrogate and override. This was precisely his case against the British government in relation to its American colonies, for whose rights against the King and the British parliament he stood resolutely. The American colonists were being denied in America the rights they would have enjoyed in Great Britain. To Burke this was totally wrong. The British government was trying to control too much of the colonists' lives, while denying them the institutional right of representation. This is hardly the attitude of a totalitarian thinker.
Burke, conservatism & open politics
Burke also believed that a coherent politics depends on traditions and institutions which make up a common culture. We may disagree with this - I think I would - but there is nothing in this view that blocks scrutiny and change. Burke holds that if a coherent politics depends on shared and entrenched traditions and institutions, those traditions and institutions must to a large extent be taken for granted but they are not sacrosanct. Any amount of change can be accommodated as long as it does not dissolve the common culture. About the permissibility and necessity of change he is explicit :
A state without the means of some change is without the means of Its conservation. Without such means it might even risk the loss of that part of the constitution
which it wished the most religiously to preserve : Reflections on the Revolution in France : https://archive.org/stream/reflectionsonthe005907mbp/reflectionsonthe005907mbp_djvu.txt.
Burke was wary of change since he believed that traditions and institutions store up the wisdom and value of the ages. We've just seen that he recognised the vital need for change at certain junctures but his sense was that traditions and (effective) institutions take time and care to build up and entrench, but are easily destroyed. Before we make changes we need to be sure that the valuable is not lost for the mere sake of the new. This is classic conservatism.
I can't read in this any idea that 'the core of the regime must not be questioned or open to inspection'. Burke saw no reason to question the common law rights of English citizens and of the principle of representation that accompanied them. That's rather a different matter from saying that they must not be questioned. And for Burke they were quite properly open to inspection - he himself inspected and explicitly defended them.
Burke as Natural Law theorist
On an endnote, some commentators see in Burke's writings traces of a Natural Law tradition, often missed because of his polemic against natural rights with which Natural Law is often confused. The issue is a tricky one. I will settle for saying that to the extent to which Burke did subscribe to a Natural Law tradition, which defines a moral standard superior to any state, government or regime, he is not a totalitarian since a totalitarian state recognises no standard or authority higher than itself. On Burke and Natural Law, see P.J. Stanlis, Edmund Burke and the natural law, University of Michigan Press, 1965.