Recently a method of doing share conversions for generic field or rings has been done here. Basically this paper offers a solution of converting a sharing of $a \in F_p$ i.e. $[a]_p$ into a sharing of $a \in F_q$ i.e. $[a]_q$.
How? Generate some random bits $[b]_p, [b]_q$ which are the same in both fields - using cut and choose. To go from $[a]_p \rightarrow [a]_q$ do the following:
- Mask your input with $\log{p}$ random bits and set $m \leftarrow \mathsf{Open}\big( [a]_p - \sum_{i=0}^{\log{p}}2^i[b_i]_p \big)$.
- Take $m$ as public input to your $F_q$ sharing algorithm and perform the subtraction modulo $p$ but this time in $F_q$ using the double shared random bits. More formally, compute $[a]_q \leftarrow \big( m + \sum_{i=0}^{\log{p}}2^i[b_i]_q \big) \bmod p$.
There is a caveat though. In order to get the right amount of statistical security and have the opened values look uniformly random you need $p$ and $q$ to be close to a power of $2$.
To answer your question, just convert $[a]_p \rightarrow [a]_q$ and then do the inversion of $[a]_q \rightarrow [a^{-1}]_q$ as you described.