Not sure where you’re coming from but I’m always happy to stump for Everest whose burgers are explicitly charbroiled and come with thousand island dressing. I was told recently they’re an offshoot of the Tops Burgers family (who are also pretty decent but largely in the Pasadena area).
Also, I haven’t tried this yet only because I keep ordering other stuff but I’m curious if anyone’s had the burger at George’s in Boyle Heights. Their burritos are pretty solid so I’d hope the burger is good.
Finally, I don’t think this meets the thousand island part of your request but I recall Yuca’s has a decent cheap burger but it’s been years since I tried one.