I dont' think God actually cared about them making the tower that could reach the lower heavens where the birds fly. If you look at the passage carefully, you'll see what the real motive of the builders was, and why this was a concern to God.
Gen. 11:3 Then they said to one another, “Come, let us make bricks and bake them thoroughly.” They had brick for stone, and they had asphalt for mortar. 4 And they said, “Come, let us build ourselves a city, and a tower whose top is in the heavens; let us make a name for ourselves, lest we be scattered abroad over the face of the whole earth.”
Papais, I think you've completely misunderstood this entire passage. The men at that time were attempting to rebel against God and stay centralized on the earth and not spread out and fill the entire land. The great city and tower were a way attract them to stay near and thus disobey God.
Then look at God's response.
Gen. 11:5 But the LORD came down to see the city and the tower which the sons of men had built. 6 And the LORD said, “Indeed the people are one and they all have one language, and this is what they begin to do; now nothing that they propose to do will be withheld from them.
God says nothing about Him being afraid they might invade His home. The notion doesn't fit with the narrative at all. Very plainly He explains that this city would be the first step in them realizing all their evil desires, and repeating the folly of the last culture that God destroyed in the flood. Thus he created the languages, forcing them to break up and scatter all over the earth. And that was exactly the result.
But I just found it a bit ironic how this passage refutes the notion that these ancients believed heaven to be a solid dome and not an open expanse. For clearly these builders understood that the sky was very high and no tower could possibly reach a height to reach to sun and stars. For even a tower going up as high as the clouds would have been impossible, let alone a theoretical dome far far above the clouds. This would have been an obvious visual inference.
But when you allow scripture to define its own terms, and not try to force fit it with other ancient beliefs, you'll see the heavens (biblically) were really an expanse which encompassed the clouds and even bird flying heights, everything makes perfect sense. Building a tower just a couple hundred feet high was well within their reach, thus proving what concept of heaven they really had.
This is yet another passage in scripture that doesn't fit with the ancient cosmologies some of these 'scholars' are trying force them into. If these ancient really believed in a solid dome way up there beyond the clouds they would have never purposed to build a tower to it. Clearly this scriptural account preceded solid dome cosmologies. My thinking is, they likely developed much later after Babel.