Yes, you can use up to 15 LiFePo4 cells in series (aka 15S) at the input of the ESC without any problems.
LiFePo4 cells definitely work all Lipo ESCs... as long as you don't exceed the max input voltage of the ESC.
Your specific ESC lists that it can handle 12S Lipo batteries. This means, it can take an input voltage is greater than >50V (>4.2V*12). This is also the reason it states that you can use up to 36 NIHM cells, which are 1.5V peak voltage (1.5v*36 = 54v). 54V is the max input voltage for this ESC.
This means, you can use 15 LiFePo4 cells in series without exceeding the input voltage rating. The peak cell voltage for LiFePo4 cells is bit over 3.5V/cell... so 3.5v* 15 = 52.5v (which is below the max rated volatge)
The internals of an ESC are switching power regulators, which output a frequency modulated AC to the motor (except sometimes cheap BECs, are linear regulator, which get super hot). The switchers compensate for any battery sag, allowing you to pretty much put in any non-alkaline cells at the input (because alkaline batteries have way too much ESR). The only way to damage the ESC is to exceed the max input voltage. Some ESC may list the actual voltage, while most just list the max number of LiPo cells that the ESC can take.