There are several avenues you can take to resolve this. The first is to log into the developer center for your iOS developer account and go to the Resolution Center and verify that the reason is correct and see if they have identified specific assets (music, artwork, code that could reasonably be protected by copyright in the US.).
There is an App Review board, but I wouldn't rush to that as it's kind of a final step and you'll want all your ducks in a row if you go that route.
If you really have no clue what parts of your app might be problematic, you can use a support incident and ask Developer Technical Support to point you to the specific items that are problematic and flagged by the reviewer. My guess is you will have a good idea and can substitute a royalty free drawing or song to resubmit the app and confirm whether the items you suspect being flagged for rejection are the actual items.
At that point, you will need to decide if you want to push things and use the assets and somehow justify why copyright material isn't in your app or get approval from the copyright owner and submit that evidence to Apple.
You could also ask the reviewer to send you the list of assets so you can start the rights check documentation. Depending on the issue they see, it could be a simple letter from your lawyer to something much more involved if Apple has had previous legal requests from copyright holders and you have run afoul of that with your recent review denial.
All of the above would be most applicable if the app contains assets that were flagged. If your app during testing consumes or downloads other copyright material, you may need to have a more direct discussion with Apple to understand if you are even going to be allowed to distribute your App through Apple's storefront.
Worst case, you can request clarification of the rejection from https://developer.apple.com/appstore/contact/
In iTunes Connect, you can provide a username and password for Apple to use to test your application:
The user name and password for a full-access account for your app. This account is used during the review process and must not expire. Include details for additional accounts in the Notes field.
However, if your intentions are to not let Apple have access to this content your app will be rejected, plain and simple.
Best Answer
Your best bet would be to contact several bands and ask for written permission to use their art in your promotion.
My expectation on reading all the terms of service of many music streaming services is that they do not grant a commercial license to any of the content they provide, so you would be scraping that content against the terms of service. To my reading this is explicitly prohibited by SoundCloud:
That alone should land you in hot water with Apple's review team.
If I were submitting an app, I would be forthright and proactive about mentioning that the artwork submitted is authorized in writing by the owners and/or properly licensed. SoundCloud also makes note that you can re-use your owned content or content that you have a license to use (uploaded content with CreativeCommons that is permissive enough for commercial reuse should do the trick if you don't have a more specific license)
If you can't do that, pay an artist to make a fake album cover - perhaps mixing elements you like and paying that artist for the rights to their work. You can upload content to soundcloud quite easily so you could control that image in a manner that Apple could rapidly determine that your company was in fact the original rights holder to the content chosen for the full screen image if not for all of the smaller thumbnails.
Fair Use is only of use in the US as a defense if you are sued in court. Expecting it to get you past Apple's policies seems like an uphill battle as you don't have a right to dispute their policies and they are fairly clear that 3.1 of https://developer.apple.com/app-store/marketing/guidelines/#photography
There is some precedent that thumbnail images are allowable under fair use doctrine by internet search engines, but that's a far stretch from you selecting one static image for promoting your work on a third party app store.
Your defense might be you are in fact "Display the app on an Apple product exactly as a customer will experience it when the app is running." but you run the risk of Apple asking you for confirmation that you have licensed that content - either from SoundCloud or the holder of the copyright on the album cover you choose for self-promotion.