My spaniel used to be terrible at picking everything up - he picked up a dead flat fish on the beach once and i ran round after him until i was so exhausted i collapsed into a heap on the sand....feeling utterly humiliated i went straight off to the pet shop and bought him a muzzle.
We never really put it on him though, as it was clear he was doing it for attention - he still does but in the house with socks, and slippers, draught excluder etc........so whenever we went for a walk after that fish incident.........i started taking a ball which he loves. Now i take 2, and he only gets the second one thrown once he has dropped the first..........he walks past tissues and sheep muck now with out bothering.
He would even rather have the balls than treats..........
So i would get something for your Lab to Carry with him, you can get little dumb bells which they can carry. And just be vigilant when walking him, as our pooch will still try and grab choc wrappers, n stuff. But he only wants them because he knows we try and get them off him - its a bit risky, but if we ignore him, he brings it to you to remind us he has it............