You're just going to get them to look at your harder if you try this. They are going to ask you why you are changing passports and what you're trying to hide.
What injustice? You were granted a visa on the basis of doing an activity. If you are not doing that activity, they are free to both deny entry and/or revoke your visa.
You must be within the jurisdiction of the embassy to apply there. That means you must PHYSICALLY be there. It specifically asks you what is your physical location, and has a question in the application for proof of your current location. Embassies are also known to randomly summon people for interviews to ensure they are actually there, and do other things.
That's your problem. A non-OA visa is ONLY available from your passport country or a country you are a resident of. You are not a resident of Laos, and therefore your application is rejected. You can get a 90 day non-O visa in Laos, but you cannot get a 1 year non-OA visa. Sounds like you might have selected the wrong options in the visa application.
they lied to you. The Thai embassy where you physically are is the one that processes your visa application. You select that one in the e-visa system. And you pay their prices.
The agency is just smoke and mirrors. He would be paying for the babysitter directly either way. With just enough going through the agency to qualify for legal employment.