Do you cook?

function should be your first thought. Is the existing backsplash permanent?

I would find a more prominent spot in the home for your personalized tiles... how about the dining room?

Use the personalized tiles on a wall in a pleasing configuration, not where they will get greasy and dirty from cooking.

Mount each tile on a square matte black board (custom) about 12-15" square and display the tiles as a triptych

No one will see them in your kitchen and they deserve something more special. (and less greasy)

If you really want to use them there, then paint your current backsplash and mount them on to the painted splash with command strips. Then you can take them down to clean, or reuse them elsewhere down the line. If you set these special tiles into a new backsplash you will never be able to use them elsewhere. What if you move, redo the kitchen? Your precious tiles will be gone.

These are the other tiles I was planning to use with the first 3 pictured. The reason I want to put them in a free standing "frame" is so that they are not permanently connected to the house.