Any trainer should be open to that, I would think. It's a pretty typical mode.
(I've expressed an interest in programming, so now my trainer gives me "homework assignments" where I write a programming rough draft, and then we talk about it, and he adjusts it as necessary.)
For core, I swear by decline sit ups and what my trainer calls "hammers." The trick with decline sit ups is that I do them on a bench that allows you to increase the angle quite a bit, so that I'm hanging at about 55*. "Hammers" are twists using a bar and a landmine, ie:
Between high angle decline sit ups and hammers, my core is ridiculously strong now. To the point where when I suggested side planks as one of my exercises, he said, "There's no point - you're too strong for that to do anything for you." I've become convinced that the core is the magic ingredient in making every day things and all our activities just so much easier. Carrying two formless sacks of dog food on my shoulder, slip-sliding against each other? I can do it. Carrying a 90 pound dumbbell (which I can't even pick up if I only use one hand) against my side? Done! (My boyfriend said not to worry about those, because he didn't want me to hurt myself trying to carry them. He should have known not to say that!) Moving a mattress and various other weirdly shaped furniture? I'm your gal! (unfortunately)
Of course, the core goes all around the torso. I swear by good mornings and deadlifts. Both of which are rewarding if done correctly, and excellent for posterior chain.