Some things can not be practised slowly, and this is one.
You can play this Alberti bass slow and soft several different ways, none of which would work to play it fast and soft. You have to start from some method that can play it fast. This is no different from an athlete learning how to do a high jump for example - you can't learn that by walking up to the take-off position and then trying to jump, you have to be running when you get there.
You are right to use forearm rotation, but too much energy is getting from your arm into the keys. Try playing this at full speed with forearm rotation, starting fff, and making a gradual diminuendo as far as you can go. Notice carefully what it feels like as you get softer. If you can feel you are making it softer by increasing the tension in your arm or hand, you are doing it wrong!
To get very soft, you need to combine the forearm rotation with "soft fingers" which absorb the energy of the rotation instead of transmitting it to the keys. Try the extreme of making the forearm rotation action, with your fingers making contact with the keys, but not pressing them enough to play the notes at all. Again, notice carefully what this feels like, and make sure you are doing in with your hand and arm relaxed.
You then need to stiffen your fingers just enough to make the notes sound, but not enough to play them loudly. Try doing the hand motion to play the complete Alberti bass (at full speed) but with only one of your fingers stiff enough to actually play the note. Do that for all the fingers individually, then put it all together.
When you first try this, the result will probably be uneven, with the notes sometimes sounding and sometimes not. But with practice, an even result will come. And in reality, if this passage is played at up to tempo, softly, and with the sustain pedal, nobody will be able to hear a few silent notes anyway.