Search the internet and decide whether you want to go for the gold standard Yuasa or for significantly less $$, an off brand. And then there is the Lithium conundrum. I've not read any horror stories about going this route and I got one. It's now almost a year old w/ no problems. Lithium will be more expensive, if you believe the mfrs you should have a special charger (which can bring the total expenditure up to about double of a Yuasa AGM) but the weight savings means the bike will accelerate faster and be easier to pickup in a tip over. (If you believe all that, I have some beach front property for sale in Arizona). I dont know if the P model takes the same battery as my '10, but it gets a Yuasa YTZ14S, 230 CCA 11.2 AH weighs 8.6 lbs. I replaced that with a Shorai, 19 AH 12V Eq, A4, Model LFX19A4-12BS12 (cost me $157). You decide. BTW, there are lots of battery threads here, you might read a few to help make up your mind about name brand vs. off brand vs. Lithium.
All that said, I would expect a P bike to require a bigger battery (though not sure where you put it since space seems to be at a premium).