Just travelled from the USA to London with my 11" iPad Pro for the first time, and at present, the experience working with Photos/Lightroom is dramatically worse in iPadOS 13.1 than in earlier versions of iOS. I'm glad that Apple is committed to making iPad a truly professional tool that can replace a laptop (since I always travel with an iPad for reading, etc), but they've really dropped the ball so far this year - they appear to have tried to do too much, and left many things half-finished. I knew about the absence of the Image Capture API beforehand, so that wasn't a surprise, but the whole system now seems broken, no matter what path I chose.
The old way of working - import into Photos, import from there into Lightroom, delete the files in Photos - was clunky, but it mostly worked reliably (though it unfortunately made it too easy to unintentionally delete files you didn't want to). I decided to try bypassing Photos this time, and just import into Files, then import from there into Lightroom. After much experimentation and frustration, I hit on the same solution as
@Denis Pagé, and opened two instances of Files, side-by-side - this finally let me drag-and-drop from my card to a folder "On My iPad". I could not get this to work reliably copying to external storage (a USB SSD attached to the OWC USB-C Travel Hub that was acting as my card reader), and eventually used Select All > Copy > Paste between Files windows to create a backup to the USB drive.
Unfortunately, that was the extent of my meaningful successes. I had hoped to import from Files into Lightroom while in Airplane Mode (to prevent syncing), go through everything on my 8-hour flight to flag my favorites, turn Airplane Mode off when I got home and let it sync overnight, then start working in Classic from there. The closest I got to success was getting 20 Sony raw files (from my 20MP RX10 IV, so not huge) imported this way, but any attempt to import more files just failed silently, even after restarting the iPad and trying again immediately after.
At some point in the process, I decided to just do things the Old Way, and imported from Files into Photos, then tried to import from there into Lightroom. Unfortunately, no matter what I did, this would also end (at best) with the unhelpful message that "XX files failed to import" in Lightroom (tried several times, with varying numbers of images), with no explanation for why. Lightroom has full access to Photos, but would not import anything from Photos - no clue as to why.
So, in the end, I resorted to importing the photos to my Mac from the USB SSD I used as a backup on the trip, having been unable to do anything meaningful with the shots before getting home. Not for the first time, I considered grabbing a MacBook Air just to use for this purpose, but that still seems wasteful for something that Apple is (hopefully) going to solve in the not-too-distant future. They sure haven't solved it so far, though.
Are you using the cloud-based Lightroom apps for iOS, Android, Windows, Mac, and Web? Then you'll love my book, 
