I've never played Battleground Italy, so the following comments are based on my just loading the scenario and taking a quick look. I see it's the turn or so after the landings.
NOTE: I think the air OB for this scenario is screwed up. Only one Patrol air group, and that's assigned to Tactical Air Force, from which it can't be transferred. Coastal Air Force, on the other hand, has 12 air groups, none of which are patrol or TB. This doesn't match the 1943 Campaign air OB. How are you going to isolate those Sicilian ports or get naval superiority around your Italy invasion zones without PA and TB air groups? Don't know why you can't upgrade the Albemarle.
Night Fighters - You don't have any Lancs and only one group of Halifaxes. I assume you won't be doing night-time bombing. You can use the NF's as daytime escorts. Load the Beaufighters with the 6x.303's, and they pack a punch.
Recon - The F-6A in Strategic AF is a joke, given it only has low-level cameras. Put them in Coastal Air Force and fly them in one AD at 12K'. I'd combine all the other recon air groups under Strategic Air Force. Let them fly 2-3 recon missions per turn, will leave 3-4 AD's free for ground attacks. Load all other recons with medium-level cameras and fly them at 24K' (except the F-9). More effective for tactical recon than high altitude. If you're playing with fog of war turned on and can't see what German units are what, fly recon AD's over a 3 hex radius area centered on Caltanisetta and another on Adrano, flying on days 1, 2, 3, and 6 of the turn.
FB-F's - Although all your US air group are optimized for fighter rather than bomber, I'd set the P-38G's, P-39Q's, and P-40N's to bomber mission (NOTE:, don't change their training now, do it after you have Sicily wrapped up and you have some turns before you invade mainland Italy. Load the P38's with 2x1000lb bombs, the P39's with 1x500lb bomb, and the P40's with 3x500lb bombs. Same with the Kittyhawk's and Hurricanes. Load the Apaches with 2x500lb bombs.
LB's. The Bostons, Blenheims, and Baltimores are pretty crappy. Swap them for Wellington's. Upgrade one of the A-20 or B-25's to B-26. Load all B-26's with 4x1000lb bombs, which are more effective than a loadout of 8x500lb bombs, in my experience. Load the B-17's and B-24's with 1000lb bombs for ground attack, again better than twice as many 500lbs bombs. Switch the Halifax loadout to the first one with the 2K, 1K, and 500lb bombs. Load the Wellingtons with as many 500lb bombs as they'll carry. Use them for Ground Attack - Interdiction the first turn, flying on days 2, 4, and 7 of the turn. After turn 1 when you can see German units, I personally believe Ground Attack - Unit is more effective than Ground Attack - Interdict, though others disagree. Transfer all the US FB's with a Bomber mission to Tactical Air Force.
TR's - Make sure they're all at rest during air phase, so they recover Morale. Use them for Freight air transfer from Tunis to your depots in Sicily.
Malta Air Command - can handle six AD's. I'd assign most of the best Spitfire FB-F's to Malta Command and have them fly the Air Superiority AD's over your two beaches, as well as one over Messina and one over Palermo.
Strategic Air Force - Fly one Ground Attack - Port against Messina to mess up Axis supply, and another against Palermo, so they can't evacuate panzer units from that port. If you get to Messina early, you want the German-held Sicilian ports wrecked so the Axis units are trapped. With your other LB groups, fly Ground Attack - Interdict on the German units you're pretty sure you're going to attack this turn.
Tactical Air Force - Set up one Ground Support AD with all the FB's with bomber mission, assign it to 15th BR Army Group so that both 7th US Army and 8th BR Army benefit.
That's my quick take on the situation. I'd love to hear what others would do.