As near as whether or not 'upon thy belly' has any symbolic meaning there is no indication that I find in the Bible to answer that.
In the passage in which this punishment was given to the Serpent, there is much that can be extracted;
Genesis 3:14 and 15 And the LORD God said unto the serpent, Because thou hast done this, thou art cursed above all cattle, and above every beast of the field; upon thy belly shalt thou go, and dust shalt thou eat all the days of thy life: And I will put enmity between thee and the woman, and between thy seed and her seed; it shall bruise thy head, and thou shalt bruise his heel.
In the statement, thou art cursed above all cattle, we are left to wonder if before that curse the Serpent had legs, which God took away or not, which seems plausible, because if the Serpent was already legless would there be any punishment in crawling on his belly.
The second part about eating dust is also part of the punishment since being low to the ground it would now be impossible for the Serpent to avoid the dust and would be forced to do so for as long as he lived.
The last part of that is where the nebulous meaning comes in:
Genesis 3:15 And I will put enmity between thee and the woman, and between thy seed and her seed; it shall bruise thy head, and thou shalt bruise his heel.
What does God mean by putting enmity between the Serpent and the woman, and between thy seed and her seed?
There are two things of importance that we must understand in this statement.
- Enmity-The quality of being an enemy; the opposite of friendship; ill will; hatred; unfriendly dispositions; malevolence. It expresses more than aversion and less than malice, and differs from displeasure in denoting a fixed or rooted hatred, whereas displeasure is more transient.
This indicates a deep rooted hate, not to the point of trying to annihilate the Serpent species, but that there will always be animosity between them.
- God said her seed and not man's seed.
Why this becomes important is because Jesus was born with the seed of Mary, but not the seed of man, and it is commonly thought to be a prophesy of the battle between Satan and Jesus for the souls of mankind.