Totally agree that lithium is a bad choice as starter “chassis” battery because of temp limitations and charging. Most bms’s would also struggle with the high current draw at startup. That said, they are a decent choice for running an audio system as long as you are running a dcdc charger and the bms has temp protection. They have much more energy density compared to lead and they can be discharged much deeper with no damage. Also, most drop in lithiums are lithium iron phosphate (LFP) which are a very safe chemistry unlike the batteries we see catching fire in teslas and consumer electronics. LFP is arguably safer than lead acid, not prone to thermal runaway like other lithium chemistries.
The op’s problem sounds more like a charging issue rather than a battery issue. Its hard to imagine that the continuous system loads exceed the output of that alternator, so I’d suspect a bad alternator or screwed up wiring.