You need to figure out what your existing charging circuit is. As eljefino mentioned, it's likely to just use a magneto pulse, rectified through a single diode. You could use a full wave bridge rectifier to gain more output efficiency, even parallel some schottky diodes, but still you aren't likely to gain enough voltage to make up for the voltage drop, if you use a filter cap on it as the supply to a regulating charger circuit that the LiFePO4 needs. You'd typically need about a 2V margin at the target current, while the likely magneto setup isn't designed for this much overhead or else it would overcharge the battery that much more during continuous use.
In other words, I like the idea to continue using a lead acid battery with a trickle charger, or use a ~120VAC float charger with a LiFePO4 instead of expecting the generator to be able to charge it.
I took a look at one of the three, manuals HF has for that Generator and it shows there is a 12VDC, unregulated output for it. That is "probably" separate magneto coils, but it is curious that they state, not to use it while using 120VAC output too. Even so, it is good for up to 8A current, so there is plenty of margin there, to use a switching boost voltage module to get enough voltage margin to power a LiFePO4 charging circuit. This is possible to do, but does not seem as practical (and is more expensive than, and more to troubleshoot now and later) as just using a 120VAC float charger for whichever battery chemistry you decide upon.