Do you know how much of your data is stored on the internal storage and how much is on the SD card? If it's on the SD card then you should be good as you don't need to send that off for the repair.
Try putting your card into someone else's device if you can, and then you can see how much data is on there.
As you've said, photos, videos and other media (downloads etc) would probably all go to the SD card rather than clogging up the internal storage, but I'm not sure about the text messages, they're probably default to the phone.
However, if you've ever 'backed up' your SMS with the stock message app or a third party one, the backup file will probably be saved to the SD card.
Your other option is to try booting it into recovery mode and changing the default USB mode from there.
Turn off the phone and take the battery out.
Put the battery back in, and hold Volume Down, and then Power. You'll boot onto a white boot menu screen.
You don't need your touchscreen to navigate this menu, just use Volume Down and Volume Up to scroll, and Power to select.
As I'm testing this at work I can't plug my phone into USB so I can't see what options it gives you, but I've seen it suggested elsewhere on the net so it might be worth a shot.
My source within HTC has given me another solution
get hold of the android source developers kit, from within that you can install android debug bridge (ADB), it's a console like the cmd command prompt where you can navigate to the folder(s) and instruct it to copy data from the device to the PC (with the phone connected in boot menu > recovery menu). You'd need to find out which sub-menus to copy, and some of the data might not be accessible or copyable, but it's an option!