His contention is that M3 (in red) assumes the truth of the argument's conclusion (that a supreme being exists).

First, I would like to take a look at these premises, and add two more("G" standing for General):

G1: If Y is X, then Y's negation is not X.

L1: If a statement is true, then its negation is not a true statement.

M1: If a property is a perfection, then its negation is not a perfection.

G2: X's entail only X's.

L2: True statements entail only true statements.

M2: Perfections entail only Perfections.

M3: Supremacy is a perfection.

Now, can we formulate an L3 and G3 in order to show that M3 does not beg the question? I think we can, in this way:

L3: Statement Z is a true statement.

G3: Z is an X.

So, we are left with the following:

G1: If Y is X, then Y's negation is not X.

G2: X's entail only X's.

G3: Z is an X.

L1: If a statement is true, then its negation is not a true statement.

L2: True statements entail only true statements.

L3: Statement Z is a true statement.

Maydole gives the following definitions:

Perfection: a property that it is necessarily better to have than not.

Supremacy: the property that a thing has if and only if it is impossible for something to be greater and impossible for there to be something else than which it is not greater.

and defends M3 thus:

M3-1: For every Z, all of the nontautological essential properties entailed by Z are perfections if and only if the property of being a Z is a perfection.

M3-2: Every nontautological essential property entailed by the property of being supreme is a perfection.

M3: The property of being supreme is a perfection.

He also writes:

...and M3 is true because it is reasonable to assume that a thing is supreme if and only if it is necessarily greater than everything else solely by virtue of having some set of perfections, making the extension of the property of being supreme identical with the intersection of the extensions of those perfections.

We could analyze M3 using the definitions of "supremacy" and "perfection" given above in the following manner:

It is necessarily better for a being that (it is impossible for something to be greater than that being) and (it is impossible for there to be something else than which it is not greater).

Notice here that "existence" is not included. Instead, Maydole argues (in appendix 2) that M1 - M3 imply that it is possible that a Supreme Being exists. He then goes on to provide a deduction showing that the conclusion of the deduction in appendix 2 entails that a supreme being exists.

Conclusion - if Maydole begs the question, it is NOT in premise M3.

[Source: Maydole, Robert. The Ontological Argument, The Blackwell Companion to Natural Theology. 2009]