Complexity of 1-in-3 SAT variant with restrictions on "unique" variables per clause
This variant of 1-in-3 SAT is polynomial time solvable. I will try to show this using the fact that 2-CNF-SAT is polynomial time solvable: wikipedia 2-SAT. (There may be better ways or more efficient algorithms for this particular problem) 

Given a special 1-in-3 sat formula $F$, do the following:


Keep the size-1 clauses
Replace any size-2 clause $(\ell_1, \ell_2)$ by formula $(\ell_1\vee \ell_2) \wedge (\neg \ell_1 \vee \neg \ell_2)$
For a size-3 clause $(\ell_1,\ell_2,u)$ where $u$ is unique to this clause: replace it by $(\neg \ell_1 \vee \neg \ell_2)$


It is easy to see that this results in a 2-CNF formula, let's call it $F'$. Use your favorite way to solve 2-SAT to find out whether $F'$ is satisfiable.

Verify that the original 1-in-3 SAT formula is satisfiable, if and only if $F'$ was satisfiable. (If this needs more explanation let me know)