It will take a bit of luck but the odds aren't that bad for seeing the aurora on a 3 day trip. While the lights may only be visible 100 days a year, as was stated, you must remember that you basically lose 120 days during the summer when it is just too light in the sky at night for them to be seen. That alone would mean you could see them 100 out of 240 nights, a much better ratio than at first glance. Last year I took out about 20 groups in Fairbanks that came on 4 day tours. ALL saw the lights while there, though one group only got about 15 minutes on their last night.
I would advise January - February when it is the coldest. That is because the really extremely low temperatures means Fairbanks is sitting under a high pressure system so the skies would be crystal clear, the first requirement for viewing the display.
I would go on your own because then you can react to changing conditions quickly, if cloudy at some viewing points it may be clear elsewhere. Good places to stay would include Aurora Borealis Lodge(may be hard to get a room)http://www.auroracabin.com/, Chena Hot Springs Resort http://www.chenahotsprings.com/, Northern Sky Lodge http://www.northernskylodge.com/. The hot springs offers activities for guests, nothing like coming out of the rock lake at -30! SkiLand at Cleary Summit offers a nicely heated building to await the appearance of the lights. http://www.skiland.org/
You should not need 4wd but I would strongly advise renting one for the added security on ice/snow covered roads. It helps having those front wheels pull you through the curves. SLOW DOWN, every winter I see many visitors off the roads and far too many on their roofs because they don't respect the roads. The roads are safe, the drivers aren't.
Like I said at the start, some luck is needed but it is a doable visit with a good chance of getting to see the lights, just realize you may be shut out. Good luck and happy travels!