I am concerned about starting it and never being able to get rid of it. A valid concern! Ivy would ruin the fence, go up the trees, over the building to the left, and attempt to take over the whole area of grass as well.

Will Morning glory grow in shade? Some shade, but they need at least 5 hours of direct sun to flower well. Easy & cheap enough to buy some seeds & find out.

Are the morning glories started from seed as invasive as the bindweed vine that people talk about. Absolutely not, especially in your zone. Bindweed is a perennial that grows back from the roots every year and morning glories drop a lot of seeds (that are easy to pull out) but the roots will be killed by the cold.

There are many more well-behaved vines, like Clematis, that grow more slowly but don't end up causing a lot of problems later. You should look for the kind that dies above ground every year because woody perennial vines and chain-link fences are not a good mix. Too much shade will severely limit your selections of vines.

You might consider other tallish shade plants that aren't vines.

Before planting anything, I would establish a border around it so it doesn't get mowed or stepped on too much, and so you don't have to try to figure out how to keep the grass neat at the bottom without harming whatever you plant there.

Your little one would really enjoy it, too. They all do. I found it's better until they are older to only take things inside and wash them before eating. Picking a cherry tomato and popping it in your mouth can send the wrong message about eating things in the yard - to a toddler. You know your kid, but it's food for thought.