Commas might only make things worse. The same can be said of dashes, which often travel in noisy pairs. Commas, in contrast, quietly occur at many places in a sentence, so you may miss the comma that signals the end of a disruption. Moreover, commas or dashes might shift the emphasis in unintended ways. Parentheses tend to play down the material, to make it clearly subordinate.
Dashes tend to emphasize the material.
The problem with our examples, however, is not how the disruption is marked, but the disruption itself. The preceding interest payment date will happen again and again, but the closing date will only happen once. The closing date is a true exception to the general rule, and consequently was placed in an exception parenthetical. But if the closing date is an exception, it should wait until after the statement of the general rule. For the first interest payment date, interest will accrue from the closing date.