Most new divers struggle with the same issue you describe and are nowhere near as fit as you are. This should tell you that they are facing the same challenges. Lead, breathing at the top of their lungs, leading to extra weight, and not being relaxed in the water, completely non-economical movement in the water, the list is long.
Now, IMHO the part you are not considering is that as both you and a fatty improve your scuba skills your ceiling is a great deal higher than theirs. Fitness and body fat aside, your challenge may be your size, but I do think you will have an edge eventually. (though it isnt a competition)
Time in the water and becoming comfortable will improve your air consumption, and typically in very short time. There isnt anything wrong with using larger tanks while you do this. I did, and in doing so, increasing my dive time, allowed me to spend the time in the water that I needed.
You may have to buy if renting large tanks isnt an option. Just dont be surprised that you eventually plateau with the weight of the tank, and find that you can go down to an 80 or 100 and have plenty of air and dive time.
When I started diving, I ended up switching to steel 100's to stay in with my buddy. I was still at 45-50 minutes, but was at 35 or so with an 80. (I was really overweight and carrying alot of lead) I lost weight, lost lead, and completed alot of dives in a short time frame. Now Im back to an 80, run a typical rec dive profile (max 90 to 100) and get on the boat at 65 minutes with 1200 psi in the tank. While your body isnt going to change as much as mine did you are going to see improvement.
All this to say... having a large tank to keep you in the water for now is a good approach. It worked for me, let me get more minutes. As much as I am not a fan of throwing equipment at a problem, I think its a good idea. Just be willing to revisit it later as you notice that you are getting out of the water with more and more gas in the tank.