Hi! I'm Eric, a Mac expert who's been paired with you through JustAnswer.com. I'm reviewing your question now, and will return w/ an answer soon!
The solution you proposed -- placing the existing "small" image on top of a white background -- would work...you would just need to make sure that the white background itself is the same dimensions as the final movie will be. For instance, if your movie's dimensions are 1920x1080, your white background should match.
As a point of clarification, too: you would need to create this white background in a photo editing program, not in iMovie itself. You might choose to use Photoshop or GIMP; either way, you'd start with a canvas size of whatever your movie dimensions are, fill that with a white background, place your existing image onto that canvas wherever you'd like it to go, and then save that entire canvas as your new image file. That resultant image file can now be imported into iMovie and will help you avoid running into the undesired black background.
Hello! Were you able to add your images to your movie without the black background? Please let me know when you can...I'm happy to keep assisting until the problem is resolved!
Thanks. I look forward to hearing from you!