possibly a question you may not have considered, it tension ... I have found that to be just as important with flats as sound, as many are able to achieve a variety of voicings with EQ ... that and the smoothness in feel are very important to me ...
I really like the Labella 760FL's that I have on one Jazz, but I wouldn't call them strictly 'warm', but a very full range flat .. there M's would probably be warmer, but also more tension ... Labella's have a good feel to me, not slippery, but smooth
DR Flats are quite warm, and a bit more 'percussive' in sound and feel ... medium tension, also very smooth
Dean Markleys have a LOT of punch, on the warmer side, and about the same tension, and also no longer available
TI's a sound of their own, least amount of tension, and the 2 sets I have had were not the smoothest, but I probably didn't give them enough time ... I have one set on a P, that wont be there long once I start experimenting I am suspecting
Pyramid Golds very nice, warm, smooth, not quite as much tension as a couple above, all in all a very pleasing string, but quite expensive
Chromes, are a fuller spectrum flat, the set I have is very smooth, tension is just about right to me
I did not care for the old style Fender 9050's at all, but I did not try the lighter guage .. felt like playing cable, and the bass I had them on had a maxed out truss rod, and they bent that like a banana, so maybe not a fair trial .. but then I used them back in the early 70's, and now I wanted to see what else was out there also ...
Keep in mind, the flats I listed were mostly all on J's at this point, so your P may be a different issue ... if I had to buy one set right now for a P, knowing what I know, I would choose Labella Deep Talkin flats, probably try the FL's first, then if I wanted more heat the S or M next ... and, I haven't even tried either of those yet, but based on what I have used from Labella, and comments I have seen from the users that I bought used strings from, I have an idea of what they are after ... The Pyramind Gold's could be also be VERY interesting on a P, as the Markleys, and the DR's and the new Fenders and Sadowsky's and GHS Precisions that I have yet to try .. Jeesh, it never ends ...good luck in your hunt ...

...JMHO's