It's definitely illegal in all 50 states to drive with a burned-out headlight, so you're looking at a legitimate traffic violation either way. The people above are right that you could get cited, but I'd push back a bit on the "it depends on the officer's mood" angle. Most jurisdictions have pretty clear policies - either they issue fix-it tickets as standard practice or they don't. If your state uses fix-it tickets for equipment violations, you'll usually get one; if not, you get a regular fine. The annoying part is that you often don't know which category your state falls into until you actually get pulled over.
The real issue is that a single burned-out headlight makes you genuinely hard to see at night or in bad weather, especially on the left side where oncoming traffic needs to spot you. I get that you're in a time crunch with your work trip, but this is one of those situations where the two-day window actually works in your favor - replacement headlights are cheap (anywhere from $15 to $100 depending on your car) and take maybe 10 minutes to swap out, sometimes less. Hit an auto parts store on your way to work and get it done before you drive anywhere. That solves both the safety problem and any potential ticket situation.