Your Bianchi looks like it's brazed with lugs so the first thing to check is that it's actually a weldable alloy.

Not sure what the best modification would be to make the frame stiffer. A double top tube might need a sleeve around the ST where you're attaching it because you might be past the end of the butting at that point.

If it's your first time with the TIG I would anticipate needing a bit of practice. Make a few non bike projects first using cheap 1.6mm wall mild steel tubing, probably square to start with. Then move on to round tubes and figure out how to mitre them and get used to welding around the curved shape. When you can do nice joints on those find an old bike frame, cut it up and weld it back together again to get used to the really thin-wall tubes. If you just start attacking the Bianchi right off the hop there's a risk of blowing a lot of holes in it. You really only get one shot at a nice TIG weld so have to have the practice.