I may have not understood the OP. Anyway, if the battery is at 12.2 (50%) and you run the converter at its 13.6 (assuming it does not go into boost) the voltage reading should spike to maybe 13.3 and then slowly climb to 13.6
The actual battery voltage would only go up a little from the 12.2 with a short time recharge.
If the converter is not charging the battery that was at 12.2 but it went up to 12.5 with the generator on, then a possible scenario for that would be a load came off the battery so it bounced back to 12.5, but the generator would not take a load off the battery unless it was taking over from an inverter that was drawing on the battery.