Jeep Wrangler Tires

[Carseler]

I was hoping to fit BF Goodrich AT 31x10.5x15 tires to my stock 2005 Jeep Wrangler Sahara. But the guys at tiger wheel say that they are too big and rub against the fender. The spare does not fit either.
What are the biggest and best tires I can fit without modification? The spare would also need to fit under the taillight and over the bumper.

What are the normal road driving and speed like with the bigger tires?

Thanks

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