Going from 20" to 17" -18" wheels - better tow capacity?

We are newbies and getting our first trailer soon – yay! We discovered our 2005 F150 Supercrew has 20” aftermarket wheels (Toyo Proxes ST II 285/50R20 ). Looking over the Ford Towing Guide, it says GCWR 14000lbs and maximum trailer weight: 8500lbs based on 17” wheels and to deduct 500lbs for 18” wheels. If I’m doing our math correctly, our current truck is actually: GCWR 12,500 and max trail weight: 7000lbs. Our trailer has dry vehicle weight: 4740 lbs and GVWR: 6160 lbs. Our truck is V8 Trinitron 5.4L but has smaller 3.55 axle so it seems it would be wise to put smaller tires to help with performance and increase towing capacity. We did tow a similar trailer with 4000 DW and it was working hard uphill. The stock OEM’s should be: 265/60R18 (which is size of our spare tire right now). We do have brake controller & weight distribution hitch already too.

Is it worthwhile changing out our wheels for smaller 17” or 18” wheels? I’m assuming it would reduce engine load, maybe improve mpg and drive smoother? If we get 17”, we’d need to replace the spare as well. Is there a big difference between them? I’ve been trying to interpret the numbers but it sounds like thinner width would also be better? I’d appreciate any input – thanks much!