If it matters, at least more days than not per week, I think I could handle two sessions per day.
Also, switched up my powerlifting style workouts with more short rest bodybuilding style workouts. No more singles. Plenty of rest pause on my last sets.
I don't think switching to a bodybuilding style workout will help your run at all. I would recommend that you keep doing the same type of weight training, but make changes in your diet and your cardio.
For one, you need to drop weight. You're fat. It's that simple. You need to cut out all sugars, fried food, and cut back hard on breads and pastas. Your cardio will go up quick for every pound you drop. It's hard to run with that weight on you just as it gets harder to do pullups as you gain weight. So you have just under a month to drop that weight, which you can easily do. What I would recommend though, is that you cut back to 2x a week instead of 3x a week since you are in crisis mode.
As far as your run goes, a 1.5 mile is pretty easy to train for. Stop worrying about the injury crap. You're not going to break an ankle or get shin splints from such a short run. I would run 3x a week and do a different type of run each day.
This is how I trained for my USMC PFT and got to a 16 minute 3 mile from nothing....
Session A
Moderate pace (light jog) - 2 miles; Keep a nice steady pace. Nothing breakneck, but move your feet faster than a walking pace. Keep track of the time it takes you. If you have to stop and rest, that's fine. You want to decrease the time each week.
Session B
Slow pace (walking) - 1 hour for distance. Walk out 30 minutes and walk back in same time. Make sure you log the distance. Each week, you want to increase the distance by as much as you can. Focus on long strides, good form and breathing.
Session C
Fast pace ("sprinting") - 10 sprints for 10 seconds each. It doesn't matter the distance. Rest at least 5 minutes in between each sprint. You're trying to really push it, but don't go 100% effort. Sprints are really taxing on your body and can cause injury. So you want to be moving, but don't act like you're on a track.
With just three weeks to train, you're not going to have amazing breakthroughs in your time, but I think a program like I wrote up would easily help you prepare for a 1.5 mile run and pass with flying colors.