Put a treadmill on the highest incline it will go on. Set the pace to a fast walk. Do not touch the handles. Crawl to your car after. Repeat. Or do what I did, be fat for a longtime. 
if grip is still a limiting factor on the farmers walk, you can toe walk with a barbell on your back. this should also help your endurance, as your calves should be able to withstand more punishment than your grip.
WTF is so wrong with doing some heavy calf raises? Put a heavy bar on your back and come up on your toes, itâs not rocket science. Since when is accessory training âfunctionalâ?
Threewhitelights has it in a nutshell.
No Offense but Harden Up and train.
My calves suck, but the must functional way to train them would be one leg at a time right? When pulling the tire, you donât flex both calves at the same time, they work unilaterally. Maybe you should try hammering away at one legged calf raise.
Start wearing Vibram Five Fingers everyday.
I donât do military presses and front raises to get better at log pressing, I log press!
If you know that your calves break down when dragging a tire with forward lean, then drag that bitch until they get stronger!
Haha, I do that shit where I walk on the front of my foot so it hits my calves.
Were huge nerds you all realize, like half of us do that.