There is no reliable yes or no answer here. Some courts do structure DWI sentences in a way that allows early release when outstanding fines and fees are paid, particularly on misdemeanor cases. The September 29 update date may be a review hearing where that determination gets made. Whether the judge will actually sign off on release at that point depends on the specific court, the judge, and how aggressively Harris County is handling repeat DWI offenses right now.
A second DWI is treated more seriously than a first, and courts have been moving in a stricter direction on DWI cases broadly. If there was any property damage or injury involved, that further reduces the likelihood of early release, regardless of whether fines are paid.
The most direct way to find out is to call the facility and ask to speak with a counselor or case manager, who can tell you whether the fines being paid is actually tied to the update date outcome. It is also worth contacting the court clerk to confirm what the September 29 date represents and whether paying the fine before then is a factor the judge will consider.
Pay the fine if you can, and make the call. Those two steps together will tell you more than any general answer can.