Hi,
The problem is ..
* when you input negative voltages you likely are beyond input voltage range.
* the high gain of the Opamp amplifies it's input offset voltage. A little input offset makes your output signal to become unsymmetric.
* with the given input signal ... the output wants to go negative. But the lack of negative power supply makes the internal circuitry to saturate. It takes some time for the circuit to get to normal - regulating output - operation.
--> for a proper output signal the Opamp needs to be continously "in regulation".
For a closed loop gain of 1000 you need a much higher (than 1000) open loop gain. How much higher depends on the expected distortion and the load conditions.
Solution:
The easiest way is to use a negative supply.
But you said you don't need the negaive signal. But do you just don't need it or do you actively need to suppress it?
For better solutions we need to know the function of the circuit, what you need to achieve...and where the signal goes to...and how it is further processed.
Klaus