Strategies for dandelion wrangling

My fiancé and I lovingly refer to our yard as the dandelion prairie. Most of them come from our next door neighbors who never once in their lives have attempted to pull weeds, resulting in some of the tallest thickest dandelions I’ve ever seen.

I ordered a garden in a box this year for the first time. I’m starting small but plan to transition our front yard a water wise garden area over the next several years. I’ve put down a thick layer of mulch in the area (about 140sq ft) already and am going to plant in early June. However, the dandelions are already starting to move in. I’ve pulled bags full of them, digging giant holes to try and get the whole root system out, I’ve sprayed pet-friendly weed killer (a necessity for our 2 dogs) but those stubborn bastards keep coming back. Any advice????