I had that problem in October 2020 during the COVID-19 emergency decree, when I moved from Ban Kok Chaiyaphum back to Bangkok. After wasting a day going to 2 offices I dumped it on a reliable visa service. I believe the fee was ฿1100.
You should be fine. Thailand and Vietnam have evisas and I believe Lao PDR does too. The Nong Khai Thailand crossing is easy. Huay Xai exiting Lao in the northwest crosses to Chiang Khong Thailand, and gets you to/from Chiang Mai nicely. In the past I've done this same itinerary.
I have a TM60 (6 month multiple entry) and have done turn arounds with no problem, carrying no luggage. Thailand doesn't generally turn away financially sound individuals.
If you do it in Vietnam you'll be there awhile waiting for it. My friend did it 6 months ago in BKK and it took 2 weeks. Request the passport with extra visa entry pages.
George Marinkovich Update: immigration rules change constantly best to visit Thai Immigration's website. ...you have to transfer your visa to the new passport at Thai immigration. It's free to do so. When your new passport is issued the old one is useless. Most countries won't allow you to enter unless you have 6 months remaining on the validity of the passport.