Hello,
that's good news. What looked like a bandwidth problem was in fact a "delayed recovery from saturation". With the benefit of hindsight one can even recognize this in your first measurement at 10kHz: the amplified signal does not really show a sine halfewave. The rising edge of the amplified signal starts with a signifcant delay of several µs due to the preceding saturation. The 10kHz signal gave you a bigger output, as it gave the amplifier more time to recover from saturation. Only the falling edge of the amplified signal really represents the amplified sine.
I agree with your assessment that you wont need a negative supply if you can ensure, that your input signal is always positive. If your input signal really gets 0 (i.e. "almost negative"), it will bring your amplifier to the edge of saturation again. It won't be as severe as in your test measurements. But it might happen, that you get some delay and distortion on the rising pulse edges.
The nice tool of Emmanuel will tell you the same: set the input cm range from 1.65 to 1.65 and the input diff range e.g. from 0 to 0.1, and you'll get a slight violation of the output voltage range at low voltages.
You can avoid this by
a) either ensuring the diff input is always positive (which you may not like, as it is driven by your sensor) or by
b) setting Vref to a small, uncritical positive voltage like e.g. 100mV. This will add 100mV to the output and will keep you in safe distance from saturation when the input reaches 0V.