What you are not looking at comparing LiFePo4 and LiPoly are the relative weights for a given capacity. The LiFePo4 is the heaviest, while LiPoly is by far the lightest. Headways and LiFePo4 still have some endearing traits, but unless you are carrying them in a cart, I would go with Li-ion or LiPoly. Even 20Ah of the much lighter LiPoly starts to feel heavy for my pedal assist ebikes. I have been using Lipoly about 7 years and it was a learning process replete w/ charred fingers, but no fires(unintenionly
I have refined the process to a point it is perfectly safe and easy, but it still requires more attention(time) than a "plug and play" chem. This is because it should be stored at a mid to mid/high state and only top charged when heading out.
LiPoly is still the best Value, but not as much as Hobbyking doesn't have it on sale much these days.
It is still the lightest
It is still the most modular allowing one to "build a pack" for the distance anticipated.
It is still the most linear in it's discharge rate, allowing a simple Voltmeter to be a pretty good "fuel gauge".
Here's a post I made the other night(which the OP never responded to);
<<<<Unfortunely, Hobbyking hasn 't been keeping lot's of LiPoly on sale like they used to, but some does go on sale now and then. You need to get on their alerts and watch(When broswering HK's awful website go straight to the last 3 battery pages). Look for the biggest bricks that will fit the space(for a reason I'll go into in a moment). The Multistar offers big bricks, 10Ah up to 20Ah, but I'm not sure they could handle the high discharge rates you mention. Maybe the new 12C stuff. I used to get the MS 6S, 10Ah bricks for $50 (free shipping) and make 12S, 20Ah packs for $200!
In the same vein, doing 12S (6S + 6S, or 4S + 4S + 4S) gives more and less expensive buying options. I run some 14S packs using the Zippy Compact 7S, 5.2Ah that were on sale for $50 ea. and 4 make a nice 14S, 10.4Ah pack.
Really, those bikes and your new packs deserve a serious charging system. Get a Mean Well HLG-320-54A, a really solid, potted CC/CV charger(it's really a power supply) that has an adjustable top charge Voltage. If you don't know, LiPoly should be stored @ 3.90V or below and only top charged before use. Not doing this will shorten the service life of the batt.s. Buy one Battery Medic per brick, that why going w/ the biggest bricks possible is way to go. For example, a 2S/2P pack would be only 4 bricks and only 4 BM's would be needed.
Using this adj. power supply and the BM's take care of the HVC and balancing, so if your LV alarm is reliable, you could lose the, often troubesome, BMS.
If you are interested in going this route, I could provide the link to the BM's and provide more details how I do it.>>>
Lastly, I would say, with Ebikes in general, but especially LiPoly, there never is any reason to run a 36 V system when a 48V one can be used. Well, maybe if the bike was purposed to be used on a boardwalk or ped path only, but in general, more performance(not just speed)can be had for no addtional money or complication. For LiPoly specifically, it is always best to have the controller provide the LVC (Low Voltage Cutoff) and 12S of LiPoly matches the common 41 or 42V LVC of a 48V controller better than 10S of LiPoly matches the LVC of a 36V controller. If fact, I have used 9S which was closer to the LVC of 31 V than 10S.