I have driven from Barahona to Santiago a couple of times and I found it absolutely exhausting. Driving in the DR is much much more stressful and tiring than driving anywhere else as you are driving for yourself and the four other cars nearest to you.
I would recommend that you get off your flight and spend the first night in Jarabacoa, there are some nice hotels and some nice walks there and it's a must-see as far as I'm concerned. Don't underestimate how long it takes to get your bags, clear immigration (with their new cameras that don't work), get to the car-hire place and then get a vehicle. Everything in DR takes a lot longer than it does in the US.
From Jarabacoa, head south to Piedra Blanca and if the new road is actually open, you will have one of the most beautiful drives of your life as it goes right through the middle of nowhere, unspoilt tropical mountains. You have time for a rest stop and can arrive relaxed in Barahona in the early afternoon. I'd recommend Villa Panoramica in Barahona, the hosts Norman and Joyce are Canadian, she cooks and they have a lot of interesting stories about life in DR.