ok so you are always at a raised input voltage, look for a precision op amp, and you should be fine, its likely the offset voltage killing your circuit, as the rail to rail would not contribute anything, providing your input and output voltage is inside there respective ranges
yes there are chips to do this, but a low offset op amp does the job much better when you have the option, (under 1mV ideally)
still that circuit is un-ideal and is better done in a full differential layout, to do as such, remove both 10M resistors, and move the inverting terminals 100K resistor to the op amps output rather than ground,