You're no more entitled to use the equipment because you are...
not fat
a committed customer
a serious runner
But what you should be doing is talking to someone in charge of the gym, explain your situation.
Your a serious customer who is paying to use the equipment for 60 continuous minutes and That is what you Expect.
Ask if it is fair for you to have a full hour block to do your long run. Explain the value you bring to the gym by being a disciplined athlete (and why that desereves and exception to the rule), likely you could be inspiring other people in the gym to train and be more committed. If the gym hopes to survive it needs more people like you, and the gym should be focused on honoring its commited/repeat customers.
Directly state the value you bring to the gym. And then personally ask for an exception to the 30 minute rule.
If that person in charge does not seem to be negotating with you, figure out what he wants. But maybe there is a 0% chance he will change his policy.
At that point you can...
start asking for benefits to comensate for your loss of percieved value. Because now this gym is worth less money to you. You want a Free locker, Free gym classes, A free month membership, free guest passes. Get something back.
Go to another gym, find their policy on treadmill use. get a quote on what'd they'd offer you to switch. Bring this information to your talk.
Avoid...
loseing your cool. You can gently threaten that you know your options with other gyms (and tell them), but you'd prefer to stay. But once you give an ultimatiom (example. 60 full minutes or i quit), you end negotioation. And if you dont want to switch gyms, dont say that.
Meet the language of the gym, expolit their needs and wants, if they sell themseleves as a serious product, tell them you make serious expectations with their product and you need full results.
You can also threaten to go online and write negative reviews of the company, tell your friends your story, create negative image in the community. (If they seek to have an online pressence, that would be signifigant.)
As long as you dont come across like a jerk, but resonable and pratcial, they will like you and meet the needs you ask. Be on their side so they want you there.