The router will completely tear up your artwork at the edges. I wouldn't suggest cutting it that way.
A router is an amazing tool and I am so glad I bought one. Like you, I had almost zero woodworking experience before starting on my CP, and recently picked up a router, jigsaw, and a shopvac. Even if you don't end up using a router and just use a hole saw (which is honestly what I would reccomend since it does take some practice to fully control a router), getting a router is a great investment. It can recess, groove, clean up ragged cuts, etc. etc. Well worth the investment.
But for cutting this 3" hole, I'd just get a hole saw with a lot of teeth in it and cut the hole slowly.