Sometimes I wonder if he is actually/really as lavish as he pretends to be.
I know that this sentence can work without either actually or really, but if we were to use one of them to make an impact on the listener, which one should it be?
It depends how you would like to impact the listener.
"Actually" denotes that something else has been said earlier, and "actually" is used to clarify what that something else is. E.g.:
I used to work at a school as a teacher. Actually, I was a teacher's assistant.
"Really" can also be used in that case. "Really" can also be used in one other way that "actually" can't, and that is for reassurance. E.g.:
I was standing there doing nothing. I was really lost.
If you want to impact our listener as to how true and how real the situation was, "really" would be used, as in the above example.
However, if you were clarifying a statement, "really" and "actually" are quite interchangeable.
really, adv. : in reality : actually
actually, adv. : in act or in fact : really
In the secondary definitions, subtle differences emerge:
1 b : truly, unquestionably —used as an intensifier
c : very 2
2 —used to emphasize an assertion
2 : in point of fact —used to suggest something unexpected
In your example sentence, there is nothing to choose between really and actually; you're wondering whether perception jibes with reality.
In the following sentences, however, really and actually could not be used interchangeably:
It's a really beautiful day.
She won't actually arrive for another hour.
The links to the definitions provide other usage examples.