I believe a Lightroom iOS developer said this is a platform limitation -- there is a limited amount of time that iOS allows apps to uploading while the app is backgrounded or the device is asleep. Also, requiring the app be awake and running bypasses a separate issue where some users were surprised when Lightroom last did try to allow background uploading. If I remember correctly, the extra data or battery usage was a negative surprise for some people.
Anyway, at least while the current behavior is in place, what I recommend:
1. plug your device into power
2. launch Lightroom
3. optional: create an all black image, and import this into Lightroom
4. let Lightroom import/sync/upload
5. use iOS controls to dim the screen as much as possible. With the generally dark UI, burn in at the screen's lowest illumination should not be a probllem.
If you are still concerned about burn-in, then do try the optional step #3 above: open the all-black image and tap the screen to hide all the Lightroom app controls & text. This should make the screen all black, and not burn any pixels.
Another idea -- use a different upload machine/method:
1. if you have a desktop computer, sign into your same account using any web browser at: https://lightroom.adobe.com
2. drag and drop image files from your computer or SD memory directly into your Lightroom account.
3. After the computer uploads images using the LR Web app, wake up your iPad and see the images in the Lightroom app that your computer has uploaded