I was on Bowen Ranch road a few weeks back and my low riding car (which has over 240,000 miles on it's original suspension) made it through fine and never bottomed out. Even though the road gets pretty rough in spots as long as you take it easy there should be no problems, I usually average about 25 mph when driving on the Bowne Ranch road and slow to a crawl to get over some of the rough spots. As usual the road from the Bowen Ranch house down to the parking area is the worst part of the trip, just take your time and pick the best path to drive through it. If you are not comfortable with driving through that rough spot you can always park your car above that section and walk on down to the hot springs trail head, it is only a couple of hundred yards away.
Also as info, if you have a front wheel drive car and ever have trouble driving up and over that rough spot because your front tires are not getting enough traction to pull you through it you can always turn your car around and back it up through that rough section. The reason to back the vehicle up is you want to get more weight on your front wheel drive tires and by backing up you have more weight pushing down on the front drive tires which gives you better traction. I have had to do this a couple of times when taking my car up some steep dirt grades while car camping up on the back roads around Big Bear, it's amazing how well it works.