Does the virtual console version of the game have a new data transfer system, or will I have to manually type the password again once I get The Lost Age?
You're going to have to type the password. The Golden Sun games on the Wii U are run on a special emulator, so it essentially becomes a GBA for it... there isn't a way in the Wii U menu to transfer all the data. Just do the short bronze password. The stats that get passed over when you use the silver one are just the extra stat points you got if you used the items like Mint, Apple and Hard Nut etc on your party, and that stupid ass long Gold code for the items... the only item really worth transferring is the Cleric's Ring, and even that isn't worth that 6 page long code. I remember I tried it once... took me well over 30 minutes to input the code and I must have made a mistake somewhere because it said it was invalid. :( I then got hold of a GBA link cable and used my sister's GBA to transfer everything that way, and ended up selling all the weapons and armor I transferred as soon as I could anyway. Every single item you have at the end of GS1 is totally outclassed and useless by the time you get Isaac's group in GS2, so there's really no need to input anything other than the Bronze code, which luckily only takes a few seconds.
Also Psynergy bestowing items like the Force Orb ARE included in the Bronze code, so you'll be able to get everything in GS2.