Unique 3SAT to Unique 1-in-3SAT
Yes, a 3-SAT formula $\phi$ can be transformed into a 1-in-3 SAT formula $\phi'$ while preserving the number of satisfying assignments.  To avoid ambiguities I will use "$\vee$" between literals of a 3-SAT clause, and commas between literals of a 1-in-3 SAT clause.



Let me preliminarily show that, given two literals $a$ and $b$, we can simulate a new type of clause $(x = a \wedge b)$ that forces the value of a new variable $x$ to be $a \wedge b$ using only 1-in-3 SAT constraints, without introducing any new solution.

Consider the cluases:
$$
(\overline{b}, c, x) \wedge
(a, c, d) \wedge
(\overline{a}, e, x) \wedge
(b, e, f)
$$

If $a=\top$, and $b=\top$, then the 2nd and 4th clauses ensures that $c=d=e=f=\bot$. The 1st and 3rd clauses then ensures that $x=\top$.

If $a=\top$, and $b=\bot$, then the 2nd clause ensures that $c=d=\bot$. The 1st clause then ensures that $x=\bot$.
The 3rd clause ensures that $e=\top$, and the 4th clause implies $f=\bot$.

The case $a=\bot$, and $b=\top$ is symmetric.

If $a=\bot$ and $b=\bot$, then the 1st and 3rd clauses imply $c=e=x=\bot$.
The 2nd and 4th clauses ensure $d=f=\top$.



I am now ready to transform a formula $\phi$ of 3SAT to a formula $\phi'$ of 1-in-3 SAT.
Consider now a clause $(a \vee b \vee c)$ of $\phi$. This can be transformed into the following equivalent 1-in-3 SAT clauses:


Add a new variable $x$ that is true iff $a$ is false and $b$ is true. This is encoded by the clause $(x = \overline{a} \wedge b)$. 
Add a new variable $y$ that is true iff $a$ is false, $b$ is false, and $c$ is true. We will need an additional variable $z$. The clause $(z = \overline{a} \wedge \overline{b})$ ensures that $z$ is true if and only if $a$ is false and $b$ is false. Then, the value of $y$ can be enforced by the clause $(y = z \wedge c)$. 
If $(a \vee b \vee c)$ is true then at least one of $a$, $b$, or $c$ is true. This means that exactly one of $a$, $x$, and $y$ is true. On the converse, if  $(a \vee b \vee c)$ is false, then at all of $a$, $x$, and $y$ are false. This shows that $(a \vee b \vee c)$  is satisfiable if and only if $(a, x, y$) is satisfiable.


We have then constructed an equivalent 1-in-3 SAT formula $\phi'$ that uses a superset of the variables of the original 3 SAT formula $\phi$.
A truth assignment to the variables of $\phi'$ satisfies $\phi'$ if and only if the assignment restricted to the variables of $\phi$ satisfies $\phi$.
Therefore, if any new solution to $\phi'$ is introduced, it must be because of the newly added variables $x$, $y$, and $z$ (one set for each clause). However, the values of these variables are completely determined by the values of the variables of $\phi$.