Voltage Noise Spectrum Measurement Problems

Since you didn't post a schematic, I'm not sure what the configuration of this circuit is, but I would be concerned about using a 1 Ohm load. Even for a CFA like ADA4870 that can output up to 1A, a 1 Ohm load may be way too low; a good way to get a rough idea of what a reasonable load would be is to look at the loads used in the datasheet (20 Ohm, 50 Ohm, and in the lower +/-5V supplies, 1.2kOhm). If the output stage is too heavily loaded, the part's noise performance and performance overall will be negatively affected, and it may not be able to hit all of the datasheet specs.

In general, when we measure noise for datasheet input noise specs/curves, we measure it at the output of the DUT and then refer it back to the input by dividing by the gain. When we're assessing the input voltage noise of a single amplifier, we usually put the part into noninverting gain (i.e. 101 or 1001) and tie IN+ to GND. With no input, any noise we see at the output must have been contributed by the part itself. We measure the gained-up noise at the output using a spectrum analyzer, and then divide by the gain to refer it back to the input.

Fig 4 on pg 3 of this app note from Renesas is a good example circuit for how we measure input current noise of a single op-amp on our datasheets: https://www.renesas.com/us/en/document/apn/an1560-making-accurate-voltage-noise-and-current-noise-measurements-operational-amplifiers-down-01hz?language=en  The main difference is the addition of a (generally) large resistor from IN+ to GND, which will convert the part's input current noise into a scaled-up input voltage noise.

If you're more interested in the overall noise of your composite system, I would set the overall voltage gain to be as low as possible to minimize voltage noise at the output. Then, I would load the output lightly (10k or 1kOhm), measure the output voltage noise, and divide by the load to get the output current noise. There will be a component of output voltage noise in that measurement result that is not due to the current noise, but hopefully by selectively gaining up the output current noise by 10k or 1k, the output voltage noise's contribution to the end result will be negligible. 

The choice of spectrum analyzer doesn't really make a difference as long as its noise floor is low enough (generally true) and it has enough bandwidth to measure the entire spectrum you're interested in.