Buy some flat wound strings.
So, in my experience, the best thing you could do for a thicker sound would be to change to flat would strings. Round and pressure wound strings have that midrange grind/clang that some folks love, and works well in some styles, but is not "fat." I use LaBella Flexible flats, and they are great. They feel a lot like rounds at the same gauge (flats can be much stiffer at equivalent gauges), focus the sound on the fundamental, and drop out all of the clang and grind. I made the switch a few years ago and it made my whole sound fall into place.
That said, some of the grind will always be there on a P-bass. That's the nature of the beast. But that's what will keep the fat sound from getting buried in the mix. The low-mid bump of the p-bass pickup brings the sound of the flats forward so that they are articulate but deep. I have a bass with a sting-ray style Bartolini and I used an early 90so EMG p-bass pickup, and both were very scooped. Neither works as well as a "typical sound" p-bass pickup. Yours might not be the best example, but the basic tonality is there. (I now have a Nordstrand p/j set and love them.)
As for the amp, I have the head version of the same amp and a single 15 cabinet I use it with. It does well in rock settings. Maybe not loud enough at times, but tone works fine. Pull the mid back a bit and roll off some highs (or all of them) because that tweeter is pretty brash. But it can get very low endy.
As for the compression issue, that amp has the "toneprint" built in effects. I know that a lot of people like the SpectraComp, but I find it super midrangey. Basically, the opposite of the direction you want. You may want to experiment with the tones of some other comps that you can load in, but many of them are going to focus on that midrange you don't want. Consider dialing the effects off until you are happy with your sound.