I have been running my Lithium system directly off my two alternators for more than a year now.
I did a whole series of tests when I swapped over to lithium
and that alternator got very hot as well, they just didn't burn it out. It would be nice to monitor the Alt temp and get an idea of what’s really going on. I like the option to turn off the alternator feed, if your on a long trip and with solar there is no need to burn diesel.
expeditionportal.com
I later swapped the 120A alternator for a 250A alternator with a standard 14.5V regulator.
We have done about 2500km with them, and no problems.
I reprogrammed my Solar controllar for Lithium as well, set them 14.3V max charge, the only thing I changed was the 240V charger, as the one I had started with a desulphination cycle which is not good for the LiFePO4 batteries, so I got a Victron on that was compatible.
I even run a LiPo starter battery, which is only connected to the starter, and it's own 90A alternator. Pretty impressed by that as it can turn the motor over really fast and weighs about 8kg only, it lives under the driver's seat now.