Maybe just upgrade massively and, as already mentioned, don't go for the cheapest option. In everyday life, a normal door is sufficient.
Since I have no idea or point of comparison: What exactly is meant by "hardware store door"? And does "massive(ly)" mean solid core board? Or actually solid wood?
SSK4 already sounds intense. How much is the surcharge from SSK 3 to SSK 4? In my apartment, there is actually 24 cm aerated concrete to the hallway and the neighboring apartment. But I have now only had SSK3 installed at the apartment entrance door to have more quiet in the apartment. I initially assumed that should be enough?