In Short
The Balrogs had their wills tied to Morgoth who, unlike Sauron, was more focused on destruction than he was on domination. This meant that while the Balrog did chase the Fellowship through Moria, it was more likely he was drawn by the power (and therefore threat) Gandalf posed going through Moria rather than desiring the Ring (as the Ring only attracts those who wish to dominate, such as Boromir, and not those who have everything, such as Tom Bombadil). The Balrogs also only served Morgoth and did not serve under Sauron.
The Balrogs were in the service of Morgoth. They were bent towards his will of total destruction. After his fall and imprisonment in the Void, they went and hid.
And in Utumno he gathered his demons about him, those spirits who first adhered to him in the days of his splendour, and became most like him in his corruption: their hearts were of fire, but they were cloaked in darkness, and terror went before them; they had whips of flame.
The Silmarillion
his Balrogs, the last of his servants that remained faithful to him
Morgoth's Ring: Annals of Aman
be seen that the wills of Orcs and Balrogs etc. are part of Melkor's power 'dispersed'. Their spirit is one of hate.
ibid: Part Five - Myths Transformed, VIII
Further, they were described as spirits of "destroying fire".
The Balrogs, of whom the whips were the chief weapons, were primeval spirits of destroying fire, chief servants of the primeval Dark Power of the First Age.
The Letters of J.R.R. Tolkien: Letter 144
Again, after the Fall of Morgoth it is said they went and hid in unreachable places in the world.
The Balrogs were destroyed, save some few that fled and hid themselves in caverns inaccessible at the roots of the earth...
The Silmarillion
In my opinion (as that's all we can take) the Balrog had no intention of domination (unlike Sauron), and therefore would not have joined Sauron. The Balrog seemed only interested in fighting Gandalf and if he had defeated Gandalf, he would've returned to Moria to continue living the same life.
Why did the Balrog chase the Fellowship? The Balrog had chased the dwarves out of Moria within a year of it's awakening and had set up a domain for itself where it could wait for Morgoth's return with other servants of Morgoth. However, when Gandalf lit wet tinder in a snowstorm he made himself known to the power within Moria.
“I have written Gandalf is here in signs that all can read from Rivendell to the mouths of Anduin.”
The Lord of the Rings: The Fellowship of the Ring - Chapter 3, The Ring Goes South
When the Fellowship first rouses the orcs, the Balrog seems to make no significant attempt to chase them. However after Gandalf again presents his power, the Balrog seems to hasten to chase the Fellowship.
“But I found myself suddenly faced by something that I have not met before. I could think of nothing to do but to try and put a shutting-spell on the door. I know many; but to do things of that kind rightly requires time, and even then the door can be broken by strength.
‘As I stood there I could hear orc-voices on the other side... Then something came into the chamber – I felt it through the door, and the orcs themselves were afraid and fell silent. It laid hold of the iron ring, and then it perceived me and my spell.
‘What it was I cannot guess, but I have never felt such a challenge. The counter-spell was terrible. It nearly broke me. For an instant the door left my control and began to open! I had to speak a word of Command. That proved too great a strain. The door burst in pieces.
I wonder if that is what they meant: that the lower levels are on fire?
“Then with a rush it leaped across the fissure.
“The dark figure streaming with fire raced towards them.”
The Lord of the Rings: The Fellowship of the Ring - Chapter 5, The Bridge of Khazad-dûm
We can see that in the Chamber of Mazarbul, the Balrog came after the initial attack of the Dwarves, likely to simply ensure there were no survivors. However, after Gandalf had again revealed to the Balrog that he was a great power. After Gandalf buries the Balrog - “All the wall gave way, and the roof of the chamber as well” - the Balrog quickly goes about setting the lower chambers on fire. Likely an attempt to trap the Fellowship. Further, when the orcs again catch up with the Fellowship, it is the Balrog that "leaps" across the flames and "races" to catch the Fellowship. He seems to be in dire need to get rid of the power that has entered his domain.
This is a contrary action to the orc chieftain, who seems truly to be drawn straight to the ring, as we see below. The Balrog doesn't show this kind of behaviour initially.
a huge orc-chieftain, almost man-high, clad in black mail from head to foot, leaped into the chamber ... With a thrust of his huge hide shield he turned Boromir’s sword and bore him backwards, throwing him to the ground. Diving under Aragorn’s blow with the speed of a striking snake he charged into the Company and thrust with his spear straight at Frodo.
The Lord of the Rings: The Fellowship of the Ring - Chapter 5, The Bridge of Khazad-dûm
The above combined with the quotes from the Silmarillion and Morgoth's Ring suggest that the Balrog would wait for Morgoth and had no intention of capturing the ring and surrendering it to Sauron or any of the 9.