Do the LDS believe in the Hypostatic Union?

Mormons do not teach that Jesus was fully God before he earned it. When he earned it isn't clear.

“I testify that with unimaginable suffering and agony at an incalculable price, the Savior earned His right to be our Redeemer, our Intermediary, our Final Judge.”

— Elder Richard G. Scott, The Atonement Can Secure Your Peace and Happiness, Ensign or Liahona, Nov. 2006, 42

The gospel of Jesus Christ teaches that man is an eternal being, made in the image and likeness of God. It also holds that man is a literal child of God and has the potential to, if faithful to divine laws and ordinances, of becoming like his heavenly parent. These truths are generally well understood by Latter-day Saints.

Less well understood, however, is the fact that God is an exalted man who once lived on an earth and underwent experiences of mortality. The Prophet Joseph Smith refers to this as 'the great secret.' (Times and Seasons 5:613[15 Aug. 1844]. See also Joseph Smith, Teachings of the Prophet Joseph Smith, p. 345.) The progression of our Father in heaven to godhood, or exaltation, was strictly in accordance with eternal principles, for he who is not able to abide the law of a celestial kingdom cannot abide a celestial glory. (D&C 88:22.)

By definition, exaltation includes the ability to procreate the family unit throughout eternity. This our Father in heaven has power to do. His marriage partner is our mother in heaven. We are their spirit children, born to them in the bonds of celestial marriage.

— Achieving a Celestial Marriage, copyright 1976, Corporation of the President of The Church of Jesus Christ of Latter-day Saints, p. 129

Affable Geek's comment that "my understanding is that Mormons reject the Credal terminology, but accept the upshot" is pretty much on the money. In my research to answer this question, it seems that the Catholic concept of the Hypostatic Union is mixed in with a lot of Trinitarian implications that LDS theology rejects. However, the basic concept of the man Jesus of Nazareth having a unique nature--that of being both a mortal man and a divine God during his life on Earth--is readily accepted.

A passage in the Book of Mormon discussing the Atonement, while it does not particularly emphasize Jesus's mortal nature, speaks directly to the issue expressed in the question above as to Jesus's nature as God while in mortality:

Alma 34: 8-16

8 And now, behold, I will testify unto you of myself that these things are true. Behold, I say unto you, that I do know that Christ shall come among the children of men, to take upon him the transgressions of his people, and that he shall atone for the sins of the world; for the Lord God hath spoken it.

9 For it is expedient that an atonement should be made; for according to the great plan of the Eternal God there must be an atonement made, or else all mankind must unavoidably perish; yea, all are hardened; yea, all are fallen and are lost, and must perish except it be through the atonement which it is expedient should be made.

10 For it is expedient that there should be a great and last sacrifice; yea, not a sacrifice of man, neither of beast, neither of any manner of fowl; for it shall not be a human sacrifice; but it must be an infinite and eternal sacrifice.

11 Now there is not any man that can sacrifice his own blood which will atone for the sins of another. Now, if a man murdereth, behold will our law, which is just, take the life of his brother? I say unto you, Nay.

12 But the law requireth the life of him who hath murdered; therefore there can be nothing which is short of an infinite atonement which will suffice for the sins of the world.

13 Therefore, it is expedient that there should be a great and last sacrifice, and then shall there be, or it is expedient there should be, a stop to the shedding of blood; then shall the law of Moses be fulfilled; yea, it shall be all fulfilled, every jot and tittle, and none shall have passed away.

14 And behold, this is the whole meaning of the law, every whit pointing to that great and last sacrifice; and that great and last sacrifice will be the Son of God, yea, infinite and eternal.

15 And thus he shall bring salvation to all those who shall believe on his name; this being the intent of this last sacrifice, to bring about the bowels of mercy, which overpowereth justice, and bringeth about means unto men that they may have faith unto repentance.

16 And thus mercy can satisfy the demands of justice, and encircles them in the arms of safety, while he that exercises no faith unto repentance is exposed to the whole law of the demands of justice; therefore only unto him that has faith unto repentance is brought about the great and eternal plan of redemption.

Here, we see Alma using the divine nature of Jesus as a part of his reasoning: if he was not divine, "infinite and eternal" and something more than an ordinary man, it would not be possible for the sacrifice that marked the end of his mortal life to provide an atonement for the sins of mankind.

In a talk given in 1983, Ezra Taft Benson, who was then the President of the Quorum of the Twelve Apostles, (he became President of the Church about two years later,) spoke of Jesus's nature as both God and Man:

Jesus Christ is the Son of God.

He came to this earth at a foreappointed time through a royal birthright that preserved His godhood. Combined in His nature were the human attributes of His mortal mother and the divine attributes and powers of His Eternal Father.

His unique heredity made Him heir to the honored title—The Only Begotten Son of God in the flesh. As the Son of God, He inherited powers and intelligence which no human ever had before or since. He was literally Immanuel, which means “God with us.” (See Matt. 1:23.)

Even though He was God’s Son sent to earth, the divine plan of the Father required that Jesus be subjected to all the difficulties and tribulations of mortality. Thus He became subject to “temptations, … hunger, thirst, and fatigue.” (Mosiah 3:7.)

To qualify as the Redeemer of all our Father’s children, Jesus had to be perfectly obedient to all the laws of God. Because He subjected Himself to the will of the Father, He grew “from grace to grace, until he received a fulness” of the Father’s power. Thus He had “all power, both in heaven and on earth.” (D&C 93:13, 17.)

So yes, it's safe to say that Latter-Day Saints do believe that Jesus retained his divine nature while living as a mortal man, and do not believe that "Jesus was, for a time, not God."