If you want to accentuate a rainbow the best thing you can do is you an adjustment brush. You can do this in Lightroom or Photoshop.
Start by boosting saturation. Then boost shadows and finally boost highlights. Set feathering quite high and reduce the flow a little. Then just brush over the rainbow itself. The high level of feathering should make it so your adjustments are subtle.
If you’ve reduced your flow you can do multiple passes to get the desired effect.
Another option is to play around with the color temperature of the rainbow using white balance.
Another thing to try is to reduce clarity all the way down on the rainbow itself thereby reducing detail behind the rainbow itself.
Just remember, don’t overdo anything. Err on the side of caution.
Of course the very best thing you can do is capture an amazing rainbow in the first place. That way you don’t even have to worry about any of this!