Our raspberries are the same way - popping up all over, several feet from the original, and not where I want them. :) Just cutting them down won't solve much. It'll just grow back because of the root system. It's just the nature of the plant.
Why not dig them up, and either replant them at the lattice, give them away to friends, or sell them for a couple bucks each? Or just consider cutting them each year as part of the plant control. They don't grow back too terribly fast, so just mowing around them tends to work well for us.
Our wild blackberries and the raspberries we planted do this - growing from the root system. But our thornless blackberries grow differently. They do expand with a root system, but it stays close by. It also starts new shoots off long limbs that end up touching the ground and developing a root system. It would work well on a lattice set-up.