I think you can figure this out to cover the time needed near the hospital (either tow dry and slow or have someone else tow it), but after that when you want to enjoy the trailer for family camping, it will likely not be dry anymore and would be at the limits of your truck.
I don't know what tires you have on your truck, but if it has "p metrics" instead of LT, you will not be happy. The soft sidewalls (nice ride) will allow too much "wiggle".
Make sure if you do this to have enough tongue weight to keep sway at a minimum.
I'll join with others to offer prayers. You are thinking ahead to allow family to be close to hospital.