As Brienne tried to push through his reforms to put the French state on better financial footing, he was constantly told that the Estates General would need to be called to enable the passing of such significant measures.
However, he appears to have tried to avoid that for a long time, trying different and increasingly less legally valid methods of pushing the reforms through.
What was the specific actual fear Brienne and Louis XVI had which caused them to try to avoid calling the Estates General for so long?
Was the fear that the Third Estate would take actions similar to the actions they ended up actually taking, leading to the French Revolution?
If not that fear, what was the actual fear?
And in the end, did Brienne and Louis XVI call the Estates General because they thought they would be able to control the Estates General to avoid what they feared passing?
The reluctance to call the Estates General was due to the ongoing power struggle between the monarchy and the parlement. There was no "anticipation" of the events that followed, although the monarchy was fearful that their power would be degraded.
To understand the calling of the Estates General, and why it was so strongly resisted by the monarchy for so long, it's far more important to understand what had happened in the 30+ years prior than it is to know anything about the French Revolution-- in fact it's probably better to forget the Revolution entirely, to avoid the notion that it was anticipated, or was inevitable.
Telling the full story would likely require going back to the wars of religion in the late 16th century-early 17th century, and the Frond rebellion during Louis XIV's regency. However the most proximate starting date of the power struggle between King & parlement was the 1750s, when Louis XV's popularity took a nosedive (for various reasons).
The parlement were the French high courts of the realm, but they did not serve a merely judicial function: they also enjoyed a unique legislative function in 'registering' royal decrees. Their political power was the only political power that was held outside the Crown. As such, they began to gain a popular conception & reputation as defenders against the encroachment of royal priviledge. They were seen-- whether rightly or wrongly-- as the only power in the kingdom who could stand against the King, who could stand against tyranny and despotism-- all words hurled at the monarchy during Louis XV's reign.
The parlement were made up of the nobility, often the more newly enobled "robe nobles," who had purchased their offices from the crown. These men revelled in their role as 'defenders against tyranny,' and certainly wished to push their own power at the expense of the crown. This was usually in pursuance of what would be a Montesqieu-esqe separation of powers, with an aristocratic body acting as a "check" on monarchical authority,
However the problem, and the reason there was such a stalemate throughout the 1780s, was due to the fact that this power was not truly 'real.' While the parlement were able to present the king with a remonstrance (a refusal to register a law, tax, edit, etc.), the King could force these through at any time-- he could compell the parlement to register his will.
So on the one hand, you have the parlement, who know that their power is tenuous at best. They could proffer a remonstrance, but it was toothless. However for the King, though he held the final trump card, the optics of constantly forcing registration of his edicts was terrible, and caused the complaints of despotism against his court to grow.
On the eve of the calling of the Estates General, then, you have a monarchy who fears that by calling this body, the parlement would be able to gain the upper hand, and would have a real place, with real power, where they could push for a growth in power. The longer the monarchy could go at it alone, the longer it could retain sole power.
There wasn't a fear that the government would be overturned, or that the Third Estate would rise up, the fear existing in 1788 was that the delicate see-saw balance of power that existed between the aristocracy of the parlement and the monarchy would be tipped to the side of the parlement if the body was called.