Let $\mathcal{V}$ be a cocomplete monoidal category which can be presented by a limit theory, so that $\mathcal{V} = \mathsf{Lex}(\mathbb{T},\text{Set})$, of course this is the case of your question.

Now, let me give the answer under the assumption that $C$ has finite $\mathcal{V}$-enriched colimits. In this case the Ind-completion is described by the formula $$\mathsf{Ind}(C) = \mathcal{V}\mathsf{Lex}(C^\circ, \mathcal{V}).$$

Now, there is a forgetful functor $$\mathcal{V}\mathsf{Lex}(C^\circ, \mathcal{V}) \to \mathsf{Lex}(C^\circ_0, \mathcal{V}), $$ where by $C^\circ_0$ we intend the underlying category of $C^\circ$. This forgetful functor is clearly faithful and conservative. I always thought it should also be monadic and recently in a private conversation Adrian Miranda sketched me a convincing argument hinting that it is both monadic and comonadic, which I will skip.

Indeed it is evident though, that having a good representation of $\mathsf{Lex}(C^\circ_0, \mathcal{V})$ could help us in understanding the situation. Now, follow the isos.

$$\mathsf{Lex}(C^\circ_0, \mathcal{V}) \cong \mathsf{Lex}(C^\circ_0, \mathsf{Lex}(\mathbb{T}, \text{Set})) \cong \mathsf{Lex}(\mathbb{T}, \mathsf{Lex}(C^\circ_0, \text{Set})) \cong \mathsf{Lex}(\mathbb{T}, \mathsf{Ind}(C_0)).$$

This means that, in the special case of additive categories, we get a representation of the additive Ind-completion of $C$ in the category internal abelian group object in the Set Ind-completion of $C_0$.

$$\mathsf{AddInd}(C) \to \mathsf{Ab}(\mathsf{Ind}(C_0)). $$

In the general case, we get $\mathbb{T}$-models in the $\mathsf{Ind}$-completion of $C_0$, of course.

In the special case of additivity, maybe (?) the reason for which the connection is tighter is that being additive is very close to be a property, more than a structure of the category $C_0$, so that already taking the Ind-completion does most of the job.

Let $\mathcal{V}$ be a cocomplete monoidal category which can be presented by a limit theory, so that $\mathcal{V} = \mathsf{Lex}(\mathbb{T},\text{Set})$, of course this is the case of your question.

Now, let me give the answer under the assumption that $C$ has finite $\mathcal{V}$-enriched colimits. In this case the Ind-completion is described by the formula $$\mathsf{Ind}(C) = \mathcal{V}\mathsf{Lex}(C^\circ, \mathcal{V}).$$

Now, there is a forgetful functor $$\mathcal{V}\mathsf{Lex}(C^\circ, \mathcal{V}) \to \mathsf{Lex}(C^\circ_0, \mathcal{V}), $$ where by $C^\circ_0$ we intend the underlying category of $C^\circ$. This forgetful functor is clearly faithful and conservative. I always thought it should also be monadic and recently in a private conversation Adrian Miranda sketched me a convincing argument hinting that it is both monadic and comonadic, which I will skip.

Indeed it is evident though, that having a good representation of $\mathsf{Lex}(C^\circ_0, \mathcal{V})$ could help us in understanding the situation. Now, follow the isos.

$$\mathsf{Lex}(C^\circ_0, \mathcal{V}) \cong \mathsf{Lex}(C^\circ_0, \mathsf{Lex}(\mathbb{T}, \text{Set})) \cong \mathsf{Lex}(\mathbb{T}, \mathsf{Lex}(C^\circ_0, \text{Set})) \cong \mathsf{Lex}(\mathbb{T}, \mathsf{Ind}(C_0)).$$

This means that, in the special case of additive categories, we get a representation of the additive Ind-completion of $C$ in the category of internal abelian group objects in the (Set) Ind-completion of $C_0$.

$$\mathsf{AddInd}(C) \to \mathsf{Ab}(\mathsf{Ind}(C_0)). $$

In the general case, we get $\mathbb{T}$-models in the $\mathsf{Ind}$-completion of $C_0$, of course.

In the special case of additivity, maybe (?) the reason for which the connection is tighter is that being additive is very close to be a property, more than a structure of the category $C_0$, so that already taking the Ind-completion does most of the job.

Let $\mathcal{V}$ be a cocomplete monoidal category which can be presented by a limit theory, so that $\mathcal{V} = \mathsf{Lex}(\mathbb{T},\text{Set})$, of course this is the case of your question.

Now, let me give the answer under the assumption that $C$ has finite $\mathcal{V}$-enriched colimits. In this case the Ind-completion is described by the formula $$\mathsf{Ind}(C) = \mathcal{V}\mathsf{Lex}(C^\circ, \mathcal{V}).$$

Now, there is a forgetful functor $$\mathcal{V}\mathsf{Lex}(C^\circ, \mathcal{V}) \to \mathsf{Lex}(C^\circ_0, \mathcal{V}), $$ where by $C^\circ_0$ we intend the underlying category of $C^\circ$. This forgetful functor is clearly faithful and conservative, and in a private conversation Adrian Miranda sketched me a convincing argument hinting that it is both monadic and comonadic, which I will skip.

Indeed it is evident though, that having a good representation of $\mathsf{Lex}(C^\circ_0, \mathcal{V})$ could help us in understanding the situation. Now, follow the isos.

$$\mathsf{Lex}(C^\circ_0, \mathcal{V}) \cong \mathsf{Lex}(C^\circ_0, \mathsf{Lex}(\mathbb{T}, \text{Set})) \cong \mathsf{Lex}(\mathbb{T}, \mathsf{Lex}(C^\circ_0, \text{Set})) \cong \mathsf{Lex}(\mathbb{T}, \mathsf{Ind}(C_0)).$$

This means that, in the special case of additive categories, we get a representation of the additive Ind-completion of $C$ in the category internal abelian group object in the Set Ind-completion of $C_0$.

$$\mathsf{AddInd}(C) \to \mathsf{Ab}(\mathsf{Ind}(C_0)). $$

In the general case, we get $\mathbb{T}$-models in the $\mathsf{Ind}$-completion of $C_0$, of course.

Let $\mathcal{V}$ be a cocomplete monoidal category which can be presented by a limit theory, so that $\mathcal{V} = \mathsf{Lex}(\mathbb{T},\text{Set})$, of course this is the case of your question.

Now, let me give the answer under the assumption that $C$ has finite $\mathcal{V}$-enriched colimits. In this case the Ind-completion is described by the formula $$\mathsf{Ind}(C) = \mathcal{V}\mathsf{Lex}(C^\circ, \mathcal{V}).$$

Now, there is a forgetful functor $$\mathcal{V}\mathsf{Lex}(C^\circ, \mathcal{V}) \to \mathsf{Lex}(C^\circ_0, \mathcal{V}), $$ where by $C^\circ_0$ we intend the underlying category of $C^\circ$. This forgetful functor is clearly faithful and conservative. I always thought it should also be monadic and recently in a private conversation Adrian Miranda sketched me a convincing argument hinting that it is both monadic and comonadic, which I will skip.

Indeed it is evident though, that having a good representation of $\mathsf{Lex}(C^\circ_0, \mathcal{V})$ could help us in understanding the situation. Now, follow the isos.

$$\mathsf{Lex}(C^\circ_0, \mathcal{V}) \cong \mathsf{Lex}(C^\circ_0, \mathsf{Lex}(\mathbb{T}, \text{Set})) \cong \mathsf{Lex}(\mathbb{T}, \mathsf{Lex}(C^\circ_0, \text{Set})) \cong \mathsf{Lex}(\mathbb{T}, \mathsf{Ind}(C_0)).$$

This means that, in the special case of additive categories, we get a representation of the additive Ind-completion of $C$ in the category internal abelian group object in the Set Ind-completion of $C_0$.

$$\mathsf{AddInd}(C) \to \mathsf{Ab}(\mathsf{Ind}(C_0)). $$

In the general case, we get $\mathbb{T}$-models in the $\mathsf{Ind}$-completion of $C_0$, of course.

In the special case of additivity, maybe (?) the reason for which the connection is tighter is that being additive is very close to be a property, more than a structure of the category $C_0$, so that already taking the Ind-completion does most of the job.