Inappoximability status of Max One in Three SAT for satisfiable instances
The most relevant paper I know is 

"The Complexity of Making Unique Choices: Approximating 1-in-k SAT" by Guruswami and Trevisan (link)

They give an algorithm for satisfiable instances which for $k=3$ would achieve a ratio of $\frac{4}{9}$, beating the random assignment. They also mention that 1-in-3 SAT is $\frac{5}{6}$-inapproximable, but I'm not sure if that applies to satisfiable instances. They cite Guruswami's Master's thesis for this.