Granite is more likely to fracture that short, and marble is softer. I'd be concerned.

I wonder if it could be cut at 3-1/2 inches, installed first and the counter butted up against it, effectively submerging some of the height behind the thinkness of the counter.

That's an interesting option, palimpsest. If not, would the 3" backsplash throw off all we are trying to achieve?

The easy way to accomplish this is to cut the backsplash from the same slab in a series of smaller sections and then seam them. When you take the one slab piece and section it, if the fabricator is good, then it still looks like one long piece. Itt's what we had to do on our full height backsplash. As one piece, it was too fragile and difficult to install. So, they cut it into sections and then just epoxied it together as each piece was installed. It looks like one piece from 2' away. You can see the seam if you get closer, but the only time you'll ever do that is to scrub off the grease from frying the chicken.

A two inch marble backsplash is possible; breezygirl put one in her kitchen. See the link below to a gw thread where she included a post about it.

Here is a link that might be useful: Breezygirl's two inch marble backsplash

I have a two inch splash, no problem for my fabricator and no problems with it so far. My marble is 2cm Calacatta. If yours is 3cm, or a soft marble, maybe that is the problem?

It depends on the kind of marble and the length of the pieces. "Veiny" marbles are prone to breaking at the veins.

Our is Vermont Montclair Danby 3 cm.

Pentimento posted my kitchen with 2" splashlette. Macro view:

I hadn't planned on going this route, but my fabricator made some mistakes in this side of the kitchen so I offered this as a compromise. My very veiny Carrara is 4cm thick, but my fabricator told me they couldn't make a small strip of marble with a stone that thick so they used a 2cm Carrara for the splashlette. I initially asked for a 1" strip, but 2" was as small as they could go to ensure stability.

I'm guessing that your Danby isn't so common that it's available in several thicknesses. Have you seen 2cm Danby in your area? Even if available, you'll have an issue over who pays for an additional slab just to get that thin splash material.