Just wanted to quickly update since I asked about applying for my 2nd SETV in Vientiane last week.
Everything went smooth and it was very easy. No need for the 3 proof ofs, as Tod said in my post. Crossed back into Thailand at Nong Khai with no issues. In fact, now that I think about it, I don't think anyone said anything to me at all. Oh no, that's not true. The Lao immigration officer did look at my US passport and saw I was born in Korea and was like "Oh! American AND Korean!" Then we talked for a minute about my mom being Korean and my dad white. We laughed and he asked me to wait at window 3. But otherwise it was handing things over, a quick glance from my passport to me, and then getting things handed back.
Oh! I also already had my applications printed and filled out and both Lao immigration and the Thai consulate took them.
Thanks to everyone who helped!