Let's start from the end, and use yet a different approach to ontology.
A second order predicate is a predicate about predicates. If you think of a predicate as being identified by the set of things about which it is true, a second-order predicate is a set of predicates with given properties -- a collection of sets of things gathered according to their characteristics, where those characteristics satisfy a given rule.
If being square is a predicate, being true of all squares is a second-order predicate. We can model it as the set of all sets containing all squares, each of which also contains all the other things that have some single aspect in common with all those squares. For instance, the set of all rectangles qualifies, as does the set of all figures that can be described with a single dimension, since every square has each of these properties.
Presumably each perfection is a predicate, a set of perfect exemplars of some criterion. So we have the set of things that are God(s), and your claim is that this set lies within every set of things which are in some way perfect. This set of Gods then lies within the intersection of all these collections of things that are perfect in each given way.
You can state that this set exists, but how can it be non-empty? God would have to both be perfectly square and perfectly round, as these are both ways of being perfect. So you need rules about what kinds of predicates can conceivably apply to God, or we have lost already.
So what are those rules? Without establishing what 'perfections' to omit from consideration, your proof is incomplete. The set of Gods exists, but may still be empty. You can start relaxing the constraints, but there is no guarantee you will ever get to non-empty set without omitting some given, relevant 'perfection'.
Thus, the proof remains incomplete and does not establish what it claims.
There is a more direct, less mathematical, approach to this proof. One can follow one of the paths that leads to the Hindu neti-neti 'not this/not that' approach to God.
In what way is existence a perfection, more than nonexistence? Does the perfect circle actually exist? Can you produce one? How about an infinite straight line? Or a perfect romantic couple? Or a perfect meal?
It seems, to observation, that what most perfect things have in common is that they can only be partially attained. They do not fully exist in reality. To the extent they exist, it is in some place or manner beyond or outside reality.
So if perfections themselves tend to have the quality of not existing or lying outside reality, how can one deduce that existence in reality has anything whatsoever in common with perfection? One should more likely presume that nonexistence or lying outside reality, the thing we find in common in most perfect things, is the proper corresponding perfect state.
This observation casts your second premise into enough doubt that one should not pursue it further.