Lighten does a good job of removing a background from one of my images since it's transparent on a black background, but I want to apply another blend mode to make it match my overall image better. Can I apply two blend modes in any way?
No. It is not possible to have more than one blending mode set per layer for existing pixels.
If you are painting, or drawing pixels onto a layer you can use the tool blending mode options before painting.
You can do this in Illustrator by creating a mask around the image and giving the layer a blend mode state while having the image with another blend mode. I have enjoyed working with multiply on the image and luminosity on the mask. I have not figured this out in Photoshop as of yet.