You're coming at this from the wrong direction if you're arguing for an invasion of Northern France in 1943. To achieve that you need to get as a minimum 2 US Armies to England ie 15+ divisions plus enough amphibious capability to land at least 3 divisions in the first wave. This needs to be started well before the fall of Tunisia if you want it to happen in 1943.
Specifically the decision must occur at the Symbol (Casablanca) confrence in January 1943. Thats about the latest in OTL a decision for stratigiclly meaning full 1943 invasion of NW Europe can be made.
Can you explain where you got twice as many Allied men in Italy, as IIRC divisional numbers were about equal?
The Germans had smaller divisions? Its been years since I studied the Italian campaign in any depth, but I do remember how complicated it is to sort out things like men & equipment. The obvious is comparing actual unit strengths vs number of units. More subtle is trying to sort out comparisons of battle and non battle casualties. In the latter case the winter weather caused a fair number of long term & permanent casualties among the men stuck out on the mountain sides. Then there were things like who could afford to lose the equipment and expend the artillery ammunition...
As has been pointed out before the Allies needed enough troops in Italy to guarantee a solid defence and threaten a breakthrough if the Germans reduced their forces.
What are the other choices? Invade Norway? Satisfy Churchill with a Balkans adventure? Or not invade the Italian mainland at all and leave the German ground forces unegaged for eight to ten months? There is only so much the deception operations can accomplish to aid the Red Army. Neither does picking off islands like Sardinia, Leros, Roades really do much for the casue. Or somehow in January 1943 convince Brooke & company a invasion of France in 1943 is practical? The decisions made at the Symbol confrence in January & then Post Husky in July/August more or less boxed the Allies into a Italian campaign.
Best use would probably have been to keep enough forces in Italy (and use them properly) to breach the Appenines after taking Rome and reach the Swiss border/ Alps where a much narrower front woyuld have been necessary.
That might be doable, if more Allied strength is committed to the Italian campaign earlier, and more aggresive actions early in September. ie: a amphibious landing near Rome in conjunction with op Giant in September. That would have been risky, but it might have also caused Hitler to panic & reverse approval of Kesselrings southern strategy, firmly ordering a retreat north of Rome.
Dragoon could have delayed to the autumn of 1944 and may not have been needed as the Germans might well have withdrawn by then.
No Dragoon means Marsailles/Toulon are not delivering supplies to the US 21st Army Group in September. October, & beyond. The failure to open the Atlantic & Channel ports to the Allied Supply ships in July, August, September, & even into November meant the southern France ports & nearly undamaged railroads were essential to sustaining three Allied Army Groups of seven armies in NW Europe by November. Since the bulk of the western ports remained in German hands, & were otherwise unusable until late 1944 12th & 21st Army Groups were receiving barely two thirds the expected supplies through the western ports in October/November 1944. The reinforcement schedule for the US Army was set back due to this and deliveries diverted to the southern French ports & from the 6th AG allowance. Without Marsailles/Toulon Ike would have been stalled post breakout in September without the material for further offensives, and without the 6th AG.
If one wants to avoid a Italian campaign as fought in OTL then you have to take a hard look at landing substantial forces on some other coast. That boils down to taking a chance on a 1943 invasion of France, either in the south or the west.