I experienced the same thing with Proraso White... I was adding a little glycerin to help it along but in the end, I had added some much better soaps/creams so I gave it to our youngest son. He only shaves around his beard so has a less challenging shave.There are a couple of things that might have contributes to your irritation.
1. The BIC chrome platinum blade is one of the sharpest available. Thus, they can cause irritation if the remainder of your shave procedure is not correct.
2. Proraso Red produces a suitable lather for many people, but it does not provide a high level of cushion/protection for those with sensitive skin. Proraso is a below average soap according to my evaluations. If you are going to try to shave ATG with BIC blade, you need to upgrade your lather. There are plenty of options available.