As noted before, not according to the Catechism of Pius X:
11 Q. Who are they who are outside the true Church?
A. Outside the true Church are: Infidels, Jews, heretics, apostates, schismatics, and the excommunicated.
12 Q. Who are infidels?
A. Infidels are those who have not been baptised and do not believe in Jesus Christ, because they either believe in and worship false gods as idolaters do, or though admitting one true God, they do not believe in the Messiah, neither as already come in the Person of Jesus Christ, nor as to come; for instance, Mohammedans and the like.
Source:
Catechism of St. Pius X | EWTN
To be honest I find this debate largely pointless. All seem to agree that Muslims have some very incorrect ideas about God and that they would fall into the "infidel" category mentioned above (most likely one would not use the term "infidel" nowadays but the substance behind it applies). Does it particularly matter at that point which of the two sub-categories they technically fall into?
As for Burke, his arguments on this point, at least those mentioned in the article, seem rather weak. It essentially amounts to saying that Islam has incorrect teachings that it attributes to God, and thus that makes it a different God... by which logic
every non-Catholic denomination does not believe in the same God as Catholics do. For example, from the article you linked:
"And the point I wanted to come to is this, I don’t believe it’s true that we’re all worshipping the same God, because the God of Islam is a governor. In other words, fundamentally Islam is, Sharia is their law, and that law, which comes from Allah, must dominate every man eventually.
“And it’s not a law that’s founded on love. To say that we all believe in love is simply not correct. And while our experience may be with individual Muslims may be one of people who are gentle and kind and so forth, we have to understand that in the end what they believe most deeply, that to which they ascribe in their hearts, demands that they govern the world. Whereas, in the Christian faith we’re taught that by the development of right reason, by sound metaphysics, and then that which leads to faith and to the light and strength that’s given by faith, we make our contribution to society also in terms of its governance, but the Church makes no pretense that it’s to govern the world, but rather that it’s to inspire and assist those who govern the world to act justly and rightly toward the citizens.”"
What he describes seems no different from what Christian theonomists suggest. Do they therefore believe in a different God?