First of all, find out if December 15th is too late to send the letters. If it's not too late, then you don't have a problem as long as the professor writes you a letter when he has more time.
Assuming the 15th is too late: Was the professor aware of the deadline when he offered to write you the letter? Did he have sufficient time to write the letter (I always erred on the side of at least a month, if not two)? If so, then he has a very real obligation to fulfill that promise. Writing letters of recommendation is not a favor that professors do for their students, it is a key part of their jobs as academics. Letters of recommendation is one of the key metrics by which institutions determine who they should accept. They help shape the landscape of the research community that the professors themselves are a part of. It is unfortunate for the professor that he has to spend a lot of his time writing a grant proposal to save his lab, but that does not excuse him from his other responsibilities.
Unfortunately, that doesn't necessarily mean that you should poke the professor every day and/or try to force the professor to write you the letter. Though he is the one at fault for failing to provide you the service that he should, you are likely the one who will suffer; either by not having a letter of recommendation, or having a haphazardly written letter that lacks attention to detail and fails to highlight your skills as a researcher. Normally I would tell you to find someone else to write you the letter, but given that you're out of time, I think your only option is to respectfully remind the professor that this is extremely important for your career, and that you require his letter of recommendation in order for your application to be considered valid. If he refuses to write it, then I would honestly consider that an ethical violation and encourage you to bring it up with your dean.
However, if you gave the professor 2 weeks notice to write the letter and/or did not make him aware of the deadline, then it may be hard to "force" him to fulfill his obligation as you did not perform your due diligence in lining up your letters early. That being said, most institutions have deadlines in December so I would assume that the professor was aware of the general timeline of when the recommendation would be due.