if my core is strong, it may support everything else
Yes, DP has been advised the same.
Pilates should be great in principle. But I just wanted to caution that Pilates teachers aren't experts in back pain necessarily, and that you shouldn't continue if you feel it's hurting you.
I would go back to the gp
I'd agree with this. You need to keep on going back. DP has been very bad at seeing the GP and his back has got worse. It's taken a lot of nagging for me, but after physio didn't help, he went back, and saw another GP who happened to be working that day, and who recognised his type of back pain and referred him to a specialist / senior physio who said the physio he'd previously been doing was no good for his back and gave him a new regime.
Finally, after years of me nagging, he's asking for a scan and the GP has agreed but now it's a case of waiting for an appointment with the right consultant who can refer him for a scan. (Even though at the beginning they said the same they said to you, it's only muscular, no need for a scan).
This has all taken years. Yesterday he was in so much pain he didn't get out of bed till 6pm. (That was a particularly bad day).
You need to keep your GP informed that the pain isn't going away, and how it's affecting your life. Ask for a scan. Ask what else they can offer you. If you're not getting anywhere with them, see one of the other GPs in your practice or elsewhere if possible.
Also, how's your weight? DP has a beer belly which really isn't helping. I wish he'd give up, but he won't and I can't make him! Do you drink regularly? If so, it could be an idea to cut down or give up, just while you're trying to get to grips with getting on top of your back issues.