Classic case of your inverter still thinking it's babysitting a grumpy old lead-acid — lithium will happily sit at 12.8V under load while your inverter's low-voltage cutoff is probably set somewhere around 11.8V for lead, so it shouldn't be tripping... unless your BMS is the one pulling the handbrake first.
Few things worth checking:
- BMS discharge current limit — what's your continuous rating? Cheap BMS units will fold under a 2kW kettle without blinking an apology
- Cable sizing — I went from lead to a pair of Fogstar Drift 200Ah in my van and my old 10mm cable was throttling everything before I even touched the inverter settings
- Inverter low-voltage cutoff — if it's a cheaper unit without adjustable thresholds, you might be fighting a losing battle
- Battery internal resistance — even lithium sags under serious surge loads, and inverters hate that brief dip on startup
What inverter have you actually got? If it's a Victron Multiplus you can dial the cutoffs in properly through VictronConnect — genuinely sorted my garden office setup when I had similar grief. Anything without config options is basically guessing in the dark.
Also worth checking whether the BMS and inverter are even communicating — some lithium setups will silently disconnect at the cell level and the inverter just sees it as a voltage collapse.
What's your battery make/model and approximate load when it cuts? Others on here have probably run into the exact same narrowboat setup.