It is possible that BMI will send a bill, eventually, if they notice. Note that this is separate from paying for a license to use the master (the recording).
You're right, since this isn't just streaming but also use in a larger work — you can't just use someone's song in what's essentially an advertisement without their permission
Kickstarter must have to deal with this shit all the time in the project videos
It apparently happens enough that they have to mention it:
MAKING IT YOURS
And remember, don’t use music, images, video, or other content that you don’t have the rights to. Using copyrighted material is almost always illegal and can lead to expensive lawsuits down the road. The easiest way to avoid copyright troubles? Either create all the content yourself or use content that is free for public use. For example, you may be able to use some Creative Commons-licensed music — per the terms of their licenses — that's available on SoundCloud. Other resources you can explore: Vimeo Music Store, Free Music Archive, and ccMixter.
