Taking advice on what bed to purchase from your doctor would be the as valuable as taking advice from
@backstreetboy on anything.
I mean what does a GP know about bed designs or even back problems? He will just tell you whatever some rep told him out the marketing brochure.
My advice(which is as valuable as every other clown on the internet)
Don't sleep on an old rundown mattress. I would normally say don't get to soft a mattress, but you weighing 65, should not make a dent in anything. So just get any well known quality brand. If you have back problems, chances are more likely its got to do with what you do every day, lifting stuff, sitting wrong on the wrong chair at your desk etc.
I used to get muscle spasms in my back, normally when I wake up in the morning and thought its the mattress or how I sleep. Sometimes so bad I had to go for a massage to rub it out. Then I stopped playing golf and never had a problem since, I never had a pain while playing, only the next day or two