My PJ requires a lot of care when I record on the (GZR) P pickup alone. I have to make sure that the tone is on 10, make sure I'm not overdriving the interface, etc. and sometimes it still sounds bloaty, especially if I'm recording too long without checking on the sound. The main thing is to record at a level you think is far too low, and to keep all the treble in the signal. Just set the gain on your interface really, really low. I saw a YouTube recording video, which may have been erroneous, where a guy was saying that too low of a level is when you can't see it in your DAW, and too high of a level is when you're actually clipping. For a problem bass, err on the low side and you'll be happier.
If you mess up, you can usually rescue the sound with fairly aggressive EQ. Boost the upper mids, cut the lower mids (and occasionally the lows), and if you have to, high pass.
I have a Parts P and a Mustang that don't require such a level of attention, and the same bass is easy-peasy if I'm playing through both pickups. All my basses have flats.
I'll also second what @Dbassmon said, with the proviso that if necessary, you can use your ears to actually high pass above the resonant frequency of your lowest string. When you play an E through most speakers, most of what you're hearing is 80hz or higher. Most Ampeg cabs start to roll off well above 40hz as well. You don't want to overdo it, but it can really save your track in a pinch.