I use the format [>=1000]#,##0,"K";General to display data labels using a K to denote thousands.

This makes something like 1,457,735 appear as 1,458K. Simple enough.

Is there a way to reduce the precision to perhaps three significant figures i.e. 1,460k?

If 1,458 were in D6, try this formula in E6

I tried using the formula with the K still attached to the end of the 1,458 but it threw
off the formula. So, if you can determine how to make the change prior to adding the K you should be good.

I would prefer to only change the display of the label. Your solution will change the actual value of the cell. I suspect it's the only solution so will probably have to go with that.