Op amp rail splitter virtual ground shifts when LED is on
You certainly have the right idea about your problem. A TL071 is completely inadequate for your purposes. Start with a datasheet. Look at figures 2 and 3, which show how much voltage the chip can put out. Notice that with a load resistor of 10k, you can get about +/- 4.5 volts for a +/- 5 volt supply, but if you drop the load to 2k, you can only get about +/- 3 volts. Your load is the equivalent of about 250 ohms at 4.5 volts, so you'll be in real trouble.

The quickest way around your problem, assuming your schematic is correct, is to boost the op amp current with a transistor, like







simulate this circuit &ndash; Schematic created using CircuitLab
EDIT - And a version of the circuit which will handle either type of load is a bit more complicated, but this







simulate this circuit
should work.