Picked up some 2x2 plain boards to give shooting sticks a try. How do you attach the leather? Picture for reference. Found an old belt that might work. 

ETA I'm thinking some small brass nails. Maybe add a little glue and clamp them together. 

My strips have been on for years, just use contact cement. I dont think you want screws, nails or staples any near where the barrel lays. If you want thicker than 3/4" legs buy a 1x6 oak board, rip 4 pcs.  1 1/2" wide and glue them together then dress the edges. I much prefer slotted sticks with a big tension knob vs. the several pivot holes choice. And I like blades vs. big spikes. Once mine are set they are set. I watch guys constantly fiddle with their stick set up and even have a helper assist. Never should be a 2 man set up!

If you have one in your area go to a Goodwill or a Salvation Army Thrift Store. They usually have a table full of belts for one or two bucks and it has become my source for good leather for small projects. Also check out their furniture because it can be sourced for a lot of good hardwood [table legs, stringers etc]

JB Weld 5 minute epoxy has been holding for 20 years.

To take the Gorilla glue concept a step further, I've glued 1/8" industrial felt to wood via Gorilla contact adhesive. It's merciless stuff - follow instructions because when the two sides touch that's it forever and ever, amen.

Years ago I had a contract for half dozen Stickley lounge chairs with leather capped arms. Stickley used hide glue but neither you or I want to go the trouble of a heated hide glue pot.

If your sticks are to be round and you want something that is easy to apply, fit and lasts forever, but isn't leather then answer is industrial heavy wall cable shrink tube. Power companies have it on their big trucks ( hint) and I'm sure it comes in various colors maybe even patterns and textures these days. In the boat building world it's the nuts for oar chafe guards and it just wears forever. They are water proof and positively will not slacken and slip plus you can apply finish to the sticks with no worry about a secondary glue/leather bond failure. Not a fashion statement but a bullet proof solution. May work on rectangular cross section sticks but I'd certainly mark off your location and round-over the edges or corners and test of course.   

If your not using round sticks then the flat rectangular cross section sticks will require glue bonding or stitching on wet leather and let shrink tight. If gluing make sure you put the rough raw side to the glue with tanned side up. Leather has a bias so you'll need to pull it to see which way is stretches. You'll probably find tanned cow hide easiest to locate with one rough side. 8oz or 1/16" - 1.2 mm is about right for thickness. You don't want leather too heavy and stiff. Stay away from vegetal died leather and stick with common chrome tanned. 

I've used 3M M-77 spray contact adhesive in my shops for years and it's great stuff. There is also the old ' Barge ',   leather, wood, etc contact cement. It is neoprene based, good stuff and remains flexible without bond loss. Might be harder to find but Amazon probably has it. Not knowing how and where you're placing your leather I can't tell you the tricks in laying down appliqués with contact cement but good luck. I might add that cork is a great material for jobs like this but it won't tolerate sharp bends or corners. 

Rick