AD8237 Output Offset Issue
I'm not saying for definite it is the TL084 but it isn't recommended for supplies as low as 4.2 volts. The data sheet recommends 5V as the minimum.

Also, the common mode input range is typically down to -12V on a +/-15V rail and this means you should not expect the device to operate correctly if the input is within 3V of the negative rail.

How does this translate to your circuit - well the negative rail is, in effect -2.4 volts at best so, the half rail resistor network is breaking this rule for the device.

Output from the TL084 is also limited to well-within the supply rails. Under extreme conditions of loading and temperature this might be +/-10V on a +/-15 rail. Clearly this will cause a significant problem when running on such a low voltage as the LiPo.