Does 'today I have begotten you' mean Christ became God's Son at a point in time rather than by eternal generation?

Here is a place where Protestants and Catholics definitely disagree. Protestants think 'today I have begotten you' applies to the time when Christ came to earth; the Catholic Church says it applies to the eternal being of the Son. Now, I admit I have a little problem with the Catholic view. I am not saying it is wrong, but grammatically and contextually, when you read Psalm 2 and its repetitions in Luke and Matthew, it does appear that the context and grammar point to the very time when Christ was begotten as a man, in other words when He came as the God-man. And that reading would make 'begotten' much more comprehensible, because who knows what 'begotten' means in eternity, when the Son is as eternal as the Father? The incarnation had a beginning: before it there was no God-man, after it there always will be, and that is like a begetting. So I think the Protestants have an argument the Catholics need to wrestle with more. I am not settling it, but that is the distinction between the two sides.