I believe it is linear. Define P(n) as the number of good-perfect matchings on n points. We have to match $u_1$ to something, n-1 choices, and it might as well be $v_2$. Now we are not allowed to connect $u_2$ to $v_1$, so let C(n) be the number of ways of completing a good-perfect matching on n points with one set of mismatched indices that cannot be connected. Similarly let D(n) be the number of ways of completing a good-perfect matching with one set of mismatched indices that are allowed to be matched. After we connect $u_2$ to $v_3$ we are allowed to connect $u_3$ to $v_1$, so we have a case of D.
The recurrences are $$P(n)=(n-1)C(n-1)$$ $$C(n)=(n-1)D(n-1)$$ $$D(n)=P(n-1)+(n-1)D(n-1)$$ started with $D(1)=1, C(1)=0, P(1)=0$ which leads to
This is A038205 in OEIS, which does not list a closed form solution. Note: I corrected an error in the expression for P(n) and the numbers in the table.