You do realize that flat tires are usually fixable, right? Do you have any reason to suspect the flat isn’t fixable? The best option would be to pay the 10-15 bucks to have the flat fixed before you leave (yes, even though you’re only going to put another 150 miles on it before you junk it). Another option would be to put a can of fix-a-flat in the tire, although the tire place will hate you because it makes an awful mess (and make sure it’s a newish can-- some of the old stuff could explode when the tire was dismounted).
It’s not like the doughnut spare is going to explode from driving it 150 miles below 50 mph, but driving 50 mph on the highway is not very safe, especially since you’re not really sure what the car is going to do in any kind of evasive maneuver. Exactly how dangerous it is probably depends on traffic and weather conditions, but in general I’d avoid it.