I am trying to prove that all r.e. sets are definable in arithmetic, where I define arithmetic to be $A:= \left\{ \varphi : \mathcal{N} \models \varphi \right\}$ for all sentences $\varphi$ in the language $\mathcal{L}(Q)$, $Q$ is Robinson arithmetic, and $\mathcal{N}$ is the standard interpretation of $\mathbb{N}$. I have done this by deducing the result as a corollary of a theorem relating the property of being r.e. and weakly representable.
$\textit{Theorem}:$ Let $T$ be a theory of $\mathcal{L}:= \left\{ +, \cdot, <, \text{Succ}, 0 \right\}$ such that $\mathcal{N} \models T$ and $Q \subseteq T$ and $T$ is recursively axiomatizable. Then for any relation $R$ over $\mathbb{N}^{n}$, TFAE:
$R$ is r.e.
$R$ is weakly represented in $T$ by a $\Sigma$-formula.
$R$ is weakly represented in $T$.
We say that $R$ is weakly represented in $T$ by some $n$-place formula $\psi(x_{1}, \ldots , x_{n})$ if for every $m_{1}, \ldots , m_{n} \in \mathbb{N}$, $R(m_{1}, \ldots , m_{n})$ holds iff $\vdash_{T} \psi(\overline{m}_{1}, \ldots , \overline{m}_{n})$. To prove the theorem, we proceed as follows: (3) $\Rightarrow$ (1), (1) $\Rightarrow$ (2).
(3) $\Rightarrow$ (1): We assume that $R$ is weakly represented in $T$ by $\psi$. Then for any $m_{1}, \ldots , m_{n} \in \mathbb{N}$, we know that $R(m_{1}, \ldots , m_{n})$ holds iff $\text{Thm}_{T}(\ulcorner \psi(\overline{m}_{1}, \ldots , \overline{m}_{n}) \urcorner )$. Thus, we obtain that $R$ is r.e. since $\text{Thm}_{T}$ is r.e. and the mapping $\alpha : \mathbb{N}^{n} \rightarrow \mathbb{N}$ which takes $(m_{1}, \ldots , m_{n}) \mapsto \ulcorner \psi(\overline{m}_{1}, \ldots , \overline{m}_{n}) \urcorner$ is recursive.
(1) $\Rightarrow$ (2): Since $R$ is r.e., we know that $R(m_{1}, \ldots, m_{n})$ holds iff there is some recursive $S(m_{1}, \ldots , m_{n}, p)$ which holds for $p \in \mathbb{N}$. Thus, $R$ is representable in $T$ by some $\Sigma$-formula $\varphi(x_{1}, \ldots , x_{n}, y)$. The $\Sigma$-formula $\psi(x_{1}, \ldots , x_{n})= \exists \, y \, \varphi(x_{1}, \ldots , x_{n}, y)$ weakly represents $R$ in $T$. This can be seen by the following: if $R(m_{1}, \ldots , m_{n})$ holds then $\vdash_{T} \psi(\overline{m}_{1}, \ldots , \overline{m}_{n})$. Conversely, if $\vdash_{T} \psi(\overline{m}_{1}, \ldots , \overline{m}_{n})$, then $\mathcal{N} \models \psi(\overline{m}_{1}, \ldots , \overline{m}_{n})$. Thus, $\mathcal{N} \models \varphi(\overline{m}_{1}, \ldots , \overline{m}_{n}, \overline{p})$ for some $p \in \mathbb{N}$. Since $\varphi$ is a $\Sigma$-formula, we have that $\vdash_{Q} \varphi(\overline{m}_{1}, \ldots , \overline{m}_{n}, \overline{p})$ and so $S(m_{1}, \ldots , m_{n}, p)$ holds and thus $R(m_{1}, \ldots , m_{n})$ holds.
Since $\vdash_{Q} \psi(\overline{m}_{1}, \ldots , \overline{m}_{n})$ iff $\mathcal{N} \models \psi(\overline{m}_{1}, \ldots , \overline{m}_{n})$ for $\Sigma$-formulas $\psi(x_{1}, \ldots , x_{n})$, we can conclude that $R$ is r.e. iff $R$ is definable in $\mathcal{N}$. Thus, we conclude that all r.e. sets are definable in arithmetic.
First, is this sufficient to get my desired result? Am I missing anything? Second, is there a more direct way to solve this? This appears to work, but is a bit roundabout. Any comments would be appreciated.
Because you are working with a complete theory $A$, there is no real reason to work with provability or with weak representability. The important fact is just that a set is r.e. if and only if it is $\Sigma^0_1$. This uses Kleene's T predicate. Once you have a $\Sigma^0_1$ formula defining the set, this trivially shows the set is definable in $A$, using that formula.
It is not true, however, that a set is r.e. if and only if it is definable in $\mathcal{N}$. The theory $A$ is not a recursively axiomatizable theory, so the theorem you mentioned in the question does not apply. For example, the set representing the halting problem is definable in arithmetic, and more generally we have that the Turing jump of any arithmetically definable set is also arithmetically definable. The collection of arithmetically definable sets includes all the sets in the arithmetical hierarchy.