Flats aren’t essential, just play the snot out of your rounds and don’t wipe them clean, they’ll dull out a lot and you’ll get close. The main thing flats offer is less string noise. But older rounds also make less noise because all the skin and sweat gravy gets in the grooves, mmmmm good. Try a small piece of foam or felt shoved under your strings at the bridge. That helps. Even though I hear you saying you don’t like it, that was a very common way to achieve the tone you are seeking. Since you’re using rounds keep the tone dialed back.
And learn to palm mute too.
The classic P Bass sound is really a simile for “any 60s song I like” as the reality outside the bounds of bass forums is that not every song was recorded with a P bass with flats and a mute and a pick. But a lot were.
Palm Mute Technique. Go