You might be better off having a traditional engagement photo shoot, then going to DL with a friend and duplicating some of the poses. People are a lot more likely to wait to pass if they see friends in a group taking a photo than a photographer trying to block them from passing.
If you want to try a shoot in the park some things you can try are going at rope drop on non MM day. If you go straight for something like Golden Horseshoe instead of Indy or Space you can get a few shots. The only problem is, once you draw attention to yourself you run the chance of people jumping up and trying to take pics too. Be on the first raft of the day to TSI. You wouldn't be able to to something like stand on the barrel bridge and take a bunch of shots, but there are corners and benches that many guests have no interest in and you can take shots between groups passing behind you. There's also the seating area near Matterhorn where the old motor boat ride was. It might not be recognizable as being in DL, but it gets very little traffic in the morning. Then there are places like the wishing well. You could snap 1-2 pics, then step aside, let people pass and let others take pics and then go in for a few more shots. It's just that holding a traditional photo shoot where your photographer takes a bunch of shots at once and tries different poses while expecting other guests to not walk behind and not get their own pics that would be a problem. You could do something similar with the stationary teacup by Mad Hatter, just snap a few shots then hop out if another guest is waiting, then wait for another turn. Then there are other things you could try like the carousel, where it's kind of expected that there would be other people in the background, so you could try for a time when there's not a lot of people, and then just open up the lens to blur them out. You could maybe try front and back elephants on Dumbo, but I doubt they're going to let a photographer turn backwards the whole time. You'd have to just sneak in a few shots every few seconds.