What you are describing sounds like too narrow an input common mode range, choose a point within your target input range outside the gain spike and make sure all your devices are in their intended region of operation (probably saturation for all of them) at that OP
Another possibility is that the systematic offset is altering the simulation results by saturating the amplifier (railing the output), plot the DC output during the sweep and see if stays within the intended output range; you can also try to measure the offset by keeping one of the inputs in the middle of the input range, sweep the other input within the target input range and measure the output voltage, an estimate of the offset is the differential input voltage when the output crosses the middle of the output range
Once you have this value you can add a source in line with your common mode source so that in the middle of the input range the output is in the middle of its output range, then you can repeat the common mode sweep to measure the gain and hopefully get a region of stable gain within your input range