The DASH plan is an exchange plan with specific guidelines (for example, how often to eat fish and nuts). Like all exchange plans, it's quite adaptable to weight loss.
Doctors are more likely to be familiar with DASH than Paleo, but the two diets are so compatible (in my opinion) that you could do one or the other or both with no problem.
I use a low-carb exchange plan, but I try to make Paleo-friendly foods for those exchanges. DASH guidelines (as I remember them) are relatively paleo-friendly, with some minor and a few major exceptions. The main disagreement (that stands out in my memory, it's been a while since I've read the DASH guidelines) is over the appropriate proportion of carbs to protein and DASH's focus on whole grains and beans (non-Paleo foods).
The thing I love best about exchange plan dieting, is that they're adaptable to any style of eating. I can "translate" almost any diet into an exchange plan. So if I learn something new (either in the science or in observing my own body) I can adapt it to my exchange plan. In losing my current 90 lbs, I've used exchange plans modeled after different diet plans including the old Weight Watchers (pre-1997), the Duke Diet, South Beach, Atkins, and Paleo- style. As my needs or beliefs change, I can quickly adapt rather than overhaul my diet.
I'm familiar with DASH only because I collect exchange plan cookbooks.
DASH is a little different than other exchange plans (but not by much).
Most exchange plan cookbooks can be used interchangeably, but DASH is a little different in that DASH uses an additional exchange group (the nut exchange, a DASH nut exchange I believe is equivalent to two or three fat exchanges on standard exchange plans) and DASH also counts the protein exchanges a little differently (I believe 1 DASH protein exchange is equal to 3 standard protein exchanges).