Well I just finished my first successful Target run, b"h!
Took about an hour, but that's still way better than running to CVS for OVs and then going to WM.
I first went to customer service where I loaded $1K bc I was nervous to try for more, so unfortunately, I don't have a data point on whether customer service can handle >$1K.
After that, I went to a cashier and was CC-rejected when trying to load $1K. Frustratingly, I called AMEX (it was an AMEX cc) and they removed the fraud alert. I told them I would be loading much more than just the $1K, and I was assured that all would be good. I was then able to load $1K (at now my 3rd register) but a subsequent $500 load was CC-rejected again. I called up even more upset and was told that by going lower than the $1K fraud amount, the system picked it up as fraud again since a fraudster would normally try to run a lesser amount if initially rejected at a higher amount. This rep was able to get all amounts pre-approved.
The 4th register of the day got the last $500 to work. And then I stepped out of line to start with my wife's card. I decided to go to the electronics dept nearby which was empty and the cashier there thought the prepaid RedCard was really cool and gladly loaded me up with 3 transactions! (We tried $2500 but it didn't work.)