How is your computer "not compatible with your iPhone 13" ? Is it 10+ yers old? What the heck would they have to do with it that would take a week
even if it was not compatible to upgrade it? Why would you need a lightening SD card reader? Just connect your iPhone to your PC by USB and move the files r through iTunes. There are multitudes of other ways (iCloud, 3rd party software, wifi, USB, etc) to move files from/to your phone without the need for an SD card.
Check out this article for a possibility:
Transfer photos and videos from your iPhone, iPad, or iPod touch - Apple Support