While doing a bit of research into this it seems that the info window will maintain a minimum size that is around 240x90. I saw quite a few references to this but I did not see any viable work arounds other than using a third party info window. It seems this is a limitation of the info window.
I do like the map and how you have set it as the background. You may be able to put an image in the info window to have it fill in any of the extra space. I have done similar things for info windows but not for the size specifically. To add in an image you would need to select the custom info window and then you can just add in your own <img tag.