Most inverters are in fact decent 3-stage chargers.. (Which is why I ask about the converter) so we will skip that.
however the control electronics, like anything made by man, CAN FAIL
So it's back to stick a volt meter (digital and one hopes calibrated) on the batteries and see what the charging voltage is.
NOTE: Most digital meters I've used were fairly well calibrated epically when new, this ranges from some 5 buck or free with coupon Horrible Freight jobs to some more expensive ($250 and up) meters.. And yes, I know how to calibrate them.