It's a sub 20 minute climb. If the TT bike is 2.5kg heavier, you would lose somewhere around 1 minute on the climb due to the extra weight iirc. Aero advantage for the TT bike will be minimal to non-existent at around 21km/h on the climb. Aero advantage on the first 3.5km flat, i could see somewhere between 20-30 seconds at best. Imho, doing the entire thing on a TT bike, the trade-off seems ludicrous. If you do a bikechange, it could be worth it if everything goes smoothly, but it's an added risk with minimal gains, maybe 5-10 seconds best case scenario.
Road bike with TT handlebars could work, but even though it will give you somewhat of a better aero position, it still won't give you anything close to TT bike aero advantages. So how much are you really gaining by it on 3.5km? Maybe 10 seconds? But then it also adds a few hundreds of grams of weight to your setup, and how much of that advantage are you going to salvage on the top? Maybe 2-5 seconds? It's something but it's not going to make you win or lose the Tour.