Having insufficient numbers of Officers to dance, impelling Lizzie to be partnered with Darcy does work here.

There could be many reasons for the lack of numbers. Assuming that the _____shire Regiment is represented by a single Battalion would provide about thirty-five officers. It might be that not all had arrived by then. In addition, it's the Headquarters (and possible a couple of the companies - there are ten at full strength) are in Meryton. The rest would be quartered in various other villages within a few miles (marching) distance.

Not all officers would be available to attend the ball. Some might be on Leave (like Wickham going to London). Others might have drawn Duty for that evening. The scattered nature of the quartering might also have made it inconvenient for some to have attended. There's also the married ones to account for - they could dance with the locals, but their wives might take up other available partners. (Nor are the last mentioned prospective "catches").