Suppose we are given a cocomplete closed symmetric monoidal stable $(\infty,1)$-category $\mathcal{C}$ with suspension $\Sigma$, and let $X \in \mathcal{C}$ be dualizable. I'd like to create a new stable cocomplete symmetric monoidal category $\mathcal{C}[\Sigma X^{-1}]$ together with a symmetric monoidal exact and continuous functor $\mathcal{C} \to \mathcal{C}[\Sigma X^{-1}]$ with the following property: For any symmetric monoidal stable category $\mathcal{D}$,the induced functor $Fun_{ex,cont}^{\otimes}(\mathcal{C}[\Sigma X^{-1}], \mathcal{D}) \to Fun_{ex,cont}^{\otimes}(\mathcal{C}, \mathcal{D})$ induces an equivalence of $Fun_{ex,cont}^{\otimes}(\mathcal{C}[\Sigma X^{-1}], \mathcal{D})$ with the full subcategory of $Fun_{ex,cont}^{\otimes}(\mathcal{C}, \mathcal{D})$ consisting of functors $\mathcal{C} \to \mathcal{D}$ that send $\Sigma X$ to an invertible object in $\mathcal{D}$. Here the notation $Fun_{ex,cont}^{\otimes}(\mathcal{C},\mathcal{D})$ denotes the category of monoidal functors that are both exact and continuous.

My idea was to localize at the set of maps $(\Sigma (\iota)) \otimes id_Y$, where $\iota$ is the map $X \otimes X^* \to \mathbb{1}$ coming from duality data for $X$. Here $Y$ ranges over all objects in $\mathcal{C}$. However, I have no idea if the resulting category would satisfy all the requirements listed above.

EDIT: Inverting $\Sigma X$ is clearly equivalent to inverting $X$.