Hi Nathan,
First off, congrats on recognizing you do not want to delete files from the card until you have backups. This is a GREAT practice to follow.
Unfortunately there is no mechanism currently in PhotoDownloader that can automatically select only the recent shoot. In Lightroom Classic, you cannot import the same image you already have imported, but you're not working with Lightroom Classic.
I can think of two options for you.
1) If you go into Advance mode in the PhotoDownloader, you can see two buttons on the bottom: one that "Checks all," the other "Unchecks all." First, click on the Uncheck All. Now, scroll to the bottom which is where the latest files are found. Click on the last file, the last image in the list. Now scroll upward until you see that first image that you took that day (the dates should be visible) and Shift-click that file. At this point, all of the images you took that day are selected. Now tap the "Check all" button and only THOSE files will be checked. You can now download only those images and past images will not be downloaded.
Process #2) (FWIW, this is what I do when I'm on holiday and I'm using my wife's laptop)
I download all of the images from that day's shoot and spend the evening working on the images until I'm done for the day. I then take an external hard drive I travel with and download all of the images and the work I did onto the hard drive. I now have two copies of the images: one on the computer and one on the hard drive. Once I've done that I know it's safe to format the card in my camera and I can go out the next day and take more photos.
Let me know if either of these work for you.