Tom and Helge,
Not sure what will be the situation where the H connector will not work.
If you have AC input connected then your loads will be directly supplied by the grid so battery can never get to fully discharged.
If there is no AC available at the inverter/charger input then it is irrelevant that inverter is fully turned OFF as there is no AC charging available anyway and if there is solar then that will charge the battery when sun is available and when battery is sufficiently charged the inverter/charger will be turned ON by the SBMS.
So just use the T sense to control the AC charging and use the H connector to protect the battery in case the battery is fully discharged.
Helge, Having Lithium battery in absorption is just not a good idea. As mentioned above I do not see any issue with using the H connector. As long as you have AC input battery can not be discharged as load is supplied by that and if you do not have an AC connection then if battery is discharged the inverter will just be turned OFF.