Viridian, no question about it, a better TT will be able to hold a better cart. That said, the Pro-ject 5.1SE can handle a better cart than the BP2 an WILL sound much better:
Phono cartridges are incredibly sensitive transducers, so they don't just convert the grooves undulations into sound. The cartridge reads the information off the lp and translates it into something the rest of the stereo system can understand and play as music.
The quality of the audio signal depends on cartridge. That is why using a good cartridge is important, as this determines the quality of the sound.
Better cart = better reproduction of music.
Your system is good enough, actually really good. Try a better cart first, then experiment with cables, isolation, mats, clamps, etc.
I also agree with Bpoletti, the Audio Technica OC-9/II will be hard to beat at its price point if you can get a step-up device or phono stage that can handle it.