I've done quite a few tiled kitchen backsplashes. I always use AcrylPro tile adhesive (available at Home Depot and elsewhere), if the individual tiles are smaller (as yours are). The stuff is just super easy to work with. It's pre-mixed, but it doesn't dry out quickly. It grips immediately, but you can move tiles around even hours later. It's got a smooth, even consistency, like creamy peanut butter, so it's easy to spread. Simple water cleanup. When it's fully cured, it's very durable and long-lasting, just like any other mastic.

So, it's more work than a self-stick mat, but not as difficult as typical tile mastic, yet it ultimately holds as well as typical tile mastic. Another option to consider.