How does Christ's hypostatic union affect the temporal nature of the Trinity? The term hypostatic union is a concise way to say that Jesus was fully God and fully Man at the same time (glossing over the details). There are a number of opinions on the precise details of this union, but I think that by and large, most would agree that the hypostatic union did not exist before Jesus was born.

This raises an interesting question. Jesus Christ is the same yesterday, today, and forever. Yet, He changed upon being born. And given that after the Resurrection, His body still had the flesh wounds, He Ascended to Heaven in that body, and is described with a body in Revelation, a reasonable assumption would be that Christ still exists in the hypostatic union and will do so for eternity.

At first glance, it seems that an atemporal God (the Son) is joined together with a temporal body (Jesus). If this is correct, then how does the Hypostatic Union affect the temporal/atemporal nature of the Trinity? If this is incorrect, then where is my error in thinking?

Sources deriving from the writings of great theologians (especially in the first few centuries) would be most appreciated, but well-supported exegesis would be fine as well.
Opposites cannot be said truly of the same thing in the same way: the divine and human things said of Christ are, of course, in opposition, suffering and incapable of suffering, for example, or dead and immortal
St. Thomas Aquinas - On the Trinity and the Incarnation Chapter 39

St. Thomas holds that Christs two natures are neither confused nor mixed. And Jesus' body, which has natural properties is subject to them.  Christ's divine nature is not subject to the things that His human nature is subject to.
I think you're thinking that they're mixed somehow and Christ's human nature (and subsequent death) brings down the Trinity or splits it up for a bit.  But that's just not contingent on the Incarnation.    If so, the Trinity would have been affected by the Incarnation as well as by the Crucifixion and It would have been made more glorious by the Transfiguration and even more glorious by the Resurrection.

Furthermore,

It is clear also from this that, although the Son is incarnate, neither the Father nor the Holy Spirit, for all that, need be incarnate, since the Incarnation did not take place by a union in the nature in which the three divine Persons are together, but in hypostasis...

The Trinity is a plurality of Persons subsisting in one nature and the Incarnation is one Person subsisting in a plurality of natures.
So, Trinity and the Incarnations require more than complicated Venn Diagrams to explain.

  +-----------------+   (Incarnation)
 +                   +       | 
 +      Father    +--+-------|---+
 +        +------+DN +--+    |    +
  +------+-------+--+    + Son HN +
         + Spirit+       +   |    +
         +               +---|---+
          +-------------+    |

DN = divine Nature
HN = Human Nature
but an even better than a crummy ASCII venn diagram is every piece of religious art depicting Jesus ever

where you see Jesus' hand with two fingers and a thumb up representing the Blessed Trinity and two fingers down representing His two natures.