Ok, if your hunting dog is so fat he can’t handle 10k at human pace you got big problems. Pups, old dogs and illness aside-- any hunting dog ought to be able to outlast you on foot by quite a margin. Think about it, how many miles does a good dog cover during the course of a six hour hunt? More than you’re going to run any time soon.
I run two dogs nearly every day. Work days I’m usually stuck in town so it’s round and round the park we go. I like to run for at least an hour-- but I’m the one quitting, the dogs will go indefinitely. Spend $20 and get a decent leash system set up, or just make one from the parts you probably have at home. I use a leash that is like a belt around my waist and then it goes to a double collar attachment. The ropes are maybe 2 feet long. Just enough for them to run side by side by my side. Any longer and they get twisted up, or try and stop and pee on everything. I have one dog on one of those harnesses that pulls under the dog’s armpits, and the other has one of those pinch type collars- on that dog it’s better than the harness thing. Just figure out what stops your dog from pulling and use it. I’ll admit the pinch collar looks mean, but it’s way better than the old “choke chains” that never stop a dog from pulling, but make you drag a choking dog around for his whole life. With that set up I can go as long as I like (up to marathon distance) and everyone’s happy. I’m happy because I have my hands free and no dog is pulling on the leash and the dogs are happy because - hey we’re out for a run!
Everyone ought to run their dogs this way, it teaches them that you’re the pack leader and it keeps everyone in shape to perform when it counts. Just remember to set some rules. For me that means, no peeing, no pulling, and only pooping at the park. Once the dog gets used to the rules of the road it’ll feel strange running without him.