Yes, it is possible to remove the physical pain of child birth and see the message of the LORD God for its deeper and more relevant aspects.
Genesis 3:16 contains two different words describing two different types of sorrow or pain:
To the woman he said, “I will greatly increase your עִצְּבוֹנֵ֣ךְ pangs (6093) in childbearing; in בְּעֶ֖צֶב pain (6089) you shall bring forth children… (Genesis 3:16 NRSV)
The difference between the two may be seen in how the two words are used elsewhere:
he named him Noah, saying, “Out of the ground that the Lord has cursed, this one shall bring us relief from our work and from the עִצְּבוֹנֵ֣ךְ toils (6093) of our hands.” (Genesis 5:29 NRSV)
Bringing a child into the world is difficult; pregnancy is difficult; the birth is a painful experience, aptly called “labor.” While this is true, there is no reason to presume the LORD God was attempting to inflict, or even describing, physical pain.
The reason is that every mother and father knows the real work (toil) begins after the birth. The infant is helpless and requires full-time care. All of this work is an addition to the daily work before the birth. Raising children is hard work and the first woman had no family or relatives to help. For the first man and woman who chose to “go it alone” making themselves gods, bringing children into the world is the type of toil described in Genesis 5:29
The second type of pain or sorrow is significantly different from the first:
Unless the LORD builds the house, They labor in vain who build it; Unless the LORD guards the city, The watchman stays awake in vain. It is vain for you to rise up early, To sit up late, To eat the bread of בְּעֶ֖צֶב sorrows (6089); For so He gives His beloved sleep. (Psalm 1271:2 NKJV)
This is the type of pain that occurs when something happens to a child. Even after the child is grown, a mother can experience sorrow or pain when her child is hurt or hurting or in trouble. Her pain is not a result of personal injury or direct physical effect: the child does not even need to be present.
In the case of the first woman, her understanding of בְּעֶ֖צֶב pain/sorrow (6089) would change when she learned of Abel's murder. No doubt her of בְּעֶ֖צֶב pain/sorrow (6089) was greater than she experienced at either birth or during the years of raising them.
After their births the mother experienced the toil of bringing children into the world. After Cain murdered Abel she experienced a completely different and more lasting type of pain. A psychologist might call this mental anguish.
It might also be considered as a spiritual condition, a consequence of being made in the image of God. That is the LORD God made sure the woman would experience the same type of pain as He would at the physical death of anything to which He had given life.
So an alternative understanding of what was spoken to the woman might be:
To the woman He said, "I will greatly multiply your toil and conceiving..." [an increase in the number of children will mean an increase in her toil of child-rearing.]
...With sorrow you will bring forth sons. [The word is most commonly translated as "sons" - in other words, it is her sons not her daughters who will bring this sorrow.]
Your desire will be for your husband and he will govern you.
Finally if the entire passage is taken as addressing her role as mother, the woman's desire could be seen as speaking to the desire for help. That is, while the LORD God commands the man to toil (6093) in the field and the woman to toil (6093) in child-rearing, the woman will desire her husband to share in both her toil and sorrow.