Given the vagaries of cyclones and changing trades dependent on the season, it would be of use to know when you're planning this trip.
Fanning Island is near enough 1000nm and almost due south of Hawaii so in the right season it should be an easy ride either way. But if a tropical low happens along, it can be really character-building.
The routing charts for the area indicate winds for most periods as being easterly to north easterly. That makes for a sleigh-ride to Fanning, and some beating going back depending on your timing.
These are VPP-derived passage plans, the first one going south in July (look for notes between the images). As you can see, you'll be tight-reaching most of of the way but some beam/broad reaching.
This is heading south in December and as you can see, it is a tight reach all the way.
This is coming back (heading north) in July, you'll be beating 27% of the way and tight-reaching the rest
And this is heading north in December with again a tight reach all the way.
As you can see, there are seasonal differences but to be fair, if this software is to be believed, they are not great.
I know that on my boat a tight reach is a good fast point of sail so based on this data, December would be my choice but then as said, it is necessary to research the tropical storm frequencies in this area. I have not done that.