According to BMI estimations, you'd have to be a least 7 feet tall for 250 to be in a healthy weight range. Sure, body frame and amount of muscle adjusts that, but, assuming you're not an outlier, then 250 isn't "just being a big guy" - it's being overweight, or possibly even obese.
I bring this up to be honest with you, not to be mean. While in that state, they physical demands of the job will be difficult for you. But the good thing about being overweight is that you can always lose weight by adopting reasonable eating habits, complemented by exercise. Just working toward the PT requirements will be a great start; and the more you work at it, the easier it gets, because you'll be getting stronger
and losing weight!
Long story short, be honest with yourself - your body is your domain, and you control it. This is all coming from a 5'10" guy who used to weigh 235.
Good luck!