Charging LiFePO4 with car alternator

Yes it will be problematic.

A LiFePO4 battery charged by a car alternator regulator will most likely fry the alternator. The sustained low resistance of the battery will pull a high current from the alternator and it will overheat.

If the alternator does not fry you will most likely wreck the battery. LiFePO4 batteries are easy to charge however they have a set of charging requirements that are not met by a system intended to charge lead acid batteries. In particular sustained overcharge or float charge will very quickly degrade the battery.

You would need a charging system designed to charge LiFePO4 and then a method of topping up your start battery, a battery to battery charger is a good idea here. This would mean changing or modifying your alternator and installing an external charge controller.