Did Jesus weep for Carthage? For Pompeii?

Not as far as we know. Jesus never seemed to care about them. Even though he wanted to encourage love for the outsider, he always treated Israel and Jerusalem differently.

And when he drew near and saw the city, he wept over it, saying, “Would that you, even you, had known on this day the things that make for peace! But now they are hidden from your eyes. For the days will come upon you, when your enemies will set up a barricade around you and surround you and hem you in on every side and tear you down to the ground, you and your children within you. And they will not leave one stone upon another in you, because you did not know the time of your visitation.”

So here are the reprobate, who go to a fate they deserve and which, in some sense they share with many outside the covenant. And yet God seems to notice them differently than the rest.

Almost like the visible church was visible to Him as well.

The yet more explicit passage in Luke 19:41-42, has given our extremists still more trouble. We are there told that Christ wept over the very men whose doom of reprobation he then pronounced. Again, the question is raised by them, If Christ felt this tender compassion for them, why did he not exert his omnipotence for their effectual calling? And their best answer seems to be, That here it was not the divine nature in Jesus that wept, but the humanity only. Now, it will readily be conceded that the divine nature was incapable of the pain of sympathetic passion, and of the agitation of grief; but we are loath to believe that this precious incident is no manifestation of the passionless, unchangeable, yet infinitely benevolent pity of the divine nature. For, first, it would impress the common Christian mind with a most painful feeling to be thus seemingly taught that holy humanity is more generous and tender than God. The humble and simple reader of the gospels had been taught by them that there was no excellence in the humanity which was not the effect and effluence of the corresponding ineffable perfection in the divinity. Second, when we hear our Lord speaking of gathering Jerusalem’s children as a hen gathereth a chicken under her wings, and then announcing the final doom of the rejected, we seem to hear the divine nature in him, at least as much as the human. And third, such interpretations, implying some degree of dissent between the two natures, are perilous, in that they obscure that vital truth, Christ the manifestation to us of the divine nature. “He is the image of the invisible God;” “He is the brightness of his glory, and express image of his substance;” “He that hath seen me hath seen the Father, and how sayest thou then, Shew us the Father?” (Col. 1:15; Heb. 1:3; John 14:9.) It is our happiness to believe that when we see Jesus weeping over lost Jerusalem, we “have seen the Father;” we have received an insight into the divine benevolence and pity (Dabney)

It is possible some hypocritical motive might make this so in some cases.  But the Confession is clear that it is not the Flesh in every case. At least some number are moved to respond by the working of the Spirit just like those who are regenerated are so moved.

That’s what “common operation” means, common to elect and non-elect alike.

Others, not elected, although they may be called by the ministry of the Word, and may have some common operations of the Spirit, yet they never truly come unto Christ, and therefore cannot be saved