Not sure you can do any of that. The backup isn't going to contain any of the apps and iOS. iOS has to be fully installed before the transfer starts. The data goes first (can't do anything while it's transferring), then it releases while the apps install in the background. I'm sure that if there's not enough available storage, it will just give a message that there's not enough free space and quit.
If you look here, there's one possibility of "Don't Transfer Apps & Data". I guess what that would do is transfer the phone settings, but not bulk data like photos and video. Not sure what good that would do.
Use the Finder or iTunes to transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support
About the only thing I could think of is maybe find a way to use an intermediary device (which you probably don't have) with enough storage capacity, delete what you want on that device, then back up after the data trimming, and then transfer from the trimmed backup. You might be able to do that with an iPad with more storage, but I'm not sure how practical that would be and whether or not everything would be properly transferred, like app data if a certain app isn't supported for iPad.
