The APC UPSs were built and designed for lead-acid batteries and their charge curves, so it’s not possible to say that they will work correctly with a battery whose charge/discharge curve could be significantly different from lead-acid.
According to the description, it has a BMS board that protects against “Overcharge” (probably overvoltage), “Over discharge”, “Over current”, “Short circuit”, and “High temp” so they should protect he batteries against common mishaps, but it does not guarantee that APCs charging circuitry will work fine because it expects to see certain voltages at the terminals during charging, and it may not see the expected values. But regardless of what happens, the BMS should also protect the battery against APCs usual battery cooking overcharge.
What could happen is that the APC will charge the batteries, but the Li-ion batteries may never reach the voltage level needed to switch to trickle charge. One way to find this out by replacing the battery and observing what the UPSs circuitry does. Another is to find out if others did this already, and how well did it work for them.
If the BMS board works as advertised, that should protect you from firey mishaps – you could buy one set, and see how it works. I don’t know if your APCs also “cooks” lead-acids like mine, but the BMS board supposed to prevent that.
BTW, the proper way to make Li-ion batteries that can replace lead-acid is to have a BMS board that in addition to the aforementioned functionalities, also mimic the lead-acid charge curve.
