I think heat branding a simple logo is much more interesting than any form of modern print. The only factors to consider is the size of your print/brand run and whether you can find someone to create a metal brand. As we tend to keep the quantity of our handmade products fairly low, creating heat branded leather patches was an option I welcomed. We have a few simple brands that blacksmiths in Nepal made to order. The actual branding itself is quite simple and doesn't take much time, although cutting the leather to shape and size can be time consuming.
I always use a simple butane gas blow torch, which can usually be found in any large hardware store. I've found it much quicker and more comfortable using a flame that is fairly small and concentrates the heat. A wooden handle or plenty of thick leather on the brand is a must, but you could still do with a pair of thick gloves. A simple flat piece of wood that I don't mind marking usually provides a firm work surface. I don't know if wood is the best backing material or not, but it works for me. Many woods also happen to take hot brands well and a heat brand is a great way for many woodcraft artisans sign their work.
Getting the most brands from the leather hide means minimising wastage, I cut the leather into strips so I only need to focus on the distance between each brand. A spare scrap of leather is good to have at hand because you'll most likely want to test how much heat is in the brand. A really hot brand wouldn't need as much time and pressure in the leather to make its mark. I regularly have to reheat the brand when branding labels in any quantity.
If you use a scalpel or any type of cutter regularly, you'll probably have a cutting mat. If not, that flat piece of wood that you don't mind marking would do just fine. Leather and wooden work surfaces dull a scapel blade fairly quickly, but they're excellent for precise cutting. Best I mention "taking care with scapels and cutters and remembering that blades can brake easily", but with hindsight I'm more likely to say "accidents will and do happen!"
If you intend to machine stitch a thick leather patch to another material, make sure your needles are suitable for the leather. The sample below has been hand stitched on to woven wool.