Here's a proof that if $\kappa$ is an uncountable regular cardinal, then the 2-category $\mathsf{Pres}_\kappa$ of locally $\kappa$-presentable categories are closed in $\mathsf{Cat}$ under PIE limits. Here the 1-morphisms are cocontinuous functors which preserve the $\kappa$-presentable objects, or equivalently have $\kappa$-accessible right adjoints. But since $\mathsf{Pres}_\kappa$ is a non-full sub-2-category, the limits in $\mathsf{Cat}$ do not necessarily continue to be limits in $\mathsf{Cat}$. But from the description below it will be clear that the inclusion $\mathsf{Pres}_\kappa \to \mathsf{Cat}$ reflects inserters, equifiers, and $\kappa$-small products.
Products
It's easy to check that locally $\kappa$-presentable categories are closed under small products. For colimits can be computed pointwise, and a generating set is given by objects whose components are $\kappa$-presentable, and initial except on a $\kappa$-small set of coordinates. If we were working with general $\kappa$-accessible categories, this would be more complicated -- I'm not sure, we might have to raise the degree of accessibility. This does not require $\kappa$ to be uncountable. But note that if the product is of size $\kappa$ or more, then generally it will not actually be a product in the 2-category $\mathsf{Pres}_\kappa$ because a functor all of whose projections preserve $\kappa$-presentable objects will generally not itself preserve $\kappa$-presentable objects.
Inserters
Now consider the inserter $Ins(F,G)$ between functors $F,G: \mathcal{C} \to \mathcal{D}$. Colimits are computed as in $\mathcal{C}$, and it's easy to see that the objects whose underlying $\mathcal{C}$-object is $\kappa$-presentable are $\kappa$-presentable. So it suffices to show that the colimit closure of these objects is all of $Ins(F,G)$ -- by Theorem 2.5.1 in Makkai-Paré, or by Lemma 3.7 in these notes by Mike Shulman and the usual strong-generator characterization of locally $\kappa$-presentable categories. Consider an object $F(c) \overset{\gamma}{\to} G(c)$ of $Ins(F,G)$. Then $c$ has some presentability rank $\lambda$. If $\lambda \leq \kappa$ we are done, otherwise $\lambda = \mu^+$ for some uncountable $\mu$, and $c$ is a $\mu$-sized colimit of $\kappa$-presentable objects (see Adámek and Rosický Remark 1.30). Let $K: I \to \mathcal{C}$ be such a diagram. There are two cases: either $\mu$ is regular, or $\mu$ is singular.
If $\mu$ is regular.
In this case, we can replace $I$ with a $\mu$-sized poset with $\mu$-small colimits and $K$ with a functor that preserves these colimits and takes values in the $\mu$-presentable objects of $\mathcal{C}$. To see this, first replace $I$ with its free completion under $\mu$-small colimits (extending $K$ by $\mu$-small-cocontinuity); then an easy induction allows one to choose a cofinal poset closed under $\mu$-small colimits. Construct a cofinal chain $(i_\alpha)_{\alpha < \mu}$ in $I$ and a natural family of morphisms $\gamma_\alpha: F K i_\alpha \to G K i_\alpha$ as follows. Enumerate the objects of $I$ as $(x_\alpha)_{\alpha < \mu}$. Let $i_\alpha^0 = \sup(\{i_\beta \mid \beta < \alpha \} \cup \{x_\alpha\})$. Because the $Fi$'s are $\mu$-presentable and $I$ is $\mu$-filtered, we may inductively choose objects $i_\alpha^n$ and morphisms $i_\alpha^n \to i_\alpha^{n+1}$ and $\gamma_\alpha^n : F K i_\alpha^n \to G K i_\alpha^{n+1}$ such that the following diagram commutes for all $\beta < \alpha$:
$\require{AMScd}
\begin{CD}
FK i_\beta @>>> FKi_\alpha^0 @>>> \cdots @>>> FKi_\alpha^n @>>> FKi_\alpha @>>> Fc \\
@V{\gamma_\beta}VV @V{\gamma_\alpha^0}VV @VVV @V{\gamma_\alpha^n}VV @V{\gamma_\alpha}VV @V{\gamma}VV \\
GK i_\beta @>>> GKi_\alpha^1 @>>> \cdots @>>> GKi_\alpha^{n+1} @>>> GKi_\alpha @>>> Gc
\end{CD}$
In the diagram, we have set $i_\alpha = \varinjlim_{n<\omega} i_\alpha^n$ and observed that becuse this colimit is preserved by $K$, $F$, and $G$, we can define $\gamma_\alpha : FKi_\alpha \to GKi_\alpha$ to be the colimit of the $\gamma_\alpha^n$'s, and this is indeed how we define $i_\alpha$ and $\gamma_\alpha$. This is where we need $\kappa$ to be uncountable -- otherwise we cannot take the colimit of this chain when $\mu = \kappa$.
Now it is easy to see that the chain $(i_\alpha)_{\alpha < \mu}$ is cofinal in $I$, and the arrows $(FKi_\alpha \overset{\gamma_\alpha}{\to} GKi_\alpha)_{\alpha < \mu}$ define a functor into $Ins(F,G)$ taking values in the $\mu$-presentable objects whose colimit is $Fc \overset{\gamma}{\to} Gc$. By induction, the $\mu$-presentable objects are in the colimit closure of the $\kappa$-presentable objects (this is where we use that $\mu$ is regular -- otherwise $\mu$-presentability is the same thing as $\lambda$-presentability!). So $Fc \overset{\gamma}{\to} Gc$ is in this colimit closure too.
If $\mu$ is singular
The basic idea is to do what we just did for a cofinal sequence of smaller regular cardinals, and take a colimit. (Interestingly, this argument seems to work for any limit cardinal $\mu$).
Choose a sequence of regular cardinals $(\mu_\alpha)$ satisfying $\kappa \leq \sum_{\beta < \alpha} \mu_\beta < \mu_\alpha < \mu$ with supremum $\mu$. Similarly to the preliminaries before, we can replace $I$ with a $\mu$-sized poset equipped with an exhaustive filtration $I_0 \subset I_1 \subset \dots \subset I$, $\cup_\alpha I_\alpha = I$ satisfying the conditions that $|I_\alpha| = \mu_\alpha$ and $I_\alpha$ is closed under $\mu_\alpha$-small colimits, which are preserved by the inclusion $I_\alpha \to I$. And we may assume that $K$ preserves $\mu$-small colimits and that $Ki$ is $\mu_\alpha$-presentable for $i \in I_\alpha$ . We construct a chain $(i_\alpha)$ with $i_\alpha \in I_\alpha$ and a natural family of morphisms $\gamma_\alpha : Fi_\alpha \to Gi_\alpha$ by taking $i_\alpha^0 = \sup (\cup_{\beta < \alpha} I_\beta)$ and performing an iterative construction as before to choose $i_\alpha$ and $\gamma_\alpha$. It is harmless to assume that $i_\alpha \in I_\alpha$ -- otherwise $i_\alpha$ first appears in $I_{\alpha'}$ for some $\alpha'>\alpha$, and we simply modify our choice of $I_\beta$ for $\alpha \leq \beta < \alpha'$ by adding in $i_\alpha$ and closing under $\mu_\beta$-filtered colimits in $I_{\alpha'}$. As before, we have defined a chain in $Ins(F,G)$ whose colimit is $Fc \overset{\gamma}{\to} Gc$. Moreover for each $\alpha$, the object $Ki_\alpha$ is $\mu_\alpha$-presentable, and $\mu_\alpha$ is a regular cardinal strictly less than $\lambda$, so by the inductive hypothesis, $Fc \overset{\gamma}{\to} Gc$ is in the colimit closure of the $\kappa$-presentable objects as desired.
Equifiers
I think that a similar argument to the inserter case will show that locally $\kappa$-presentable categories are closed under equifiers for uncountable $\kappa$.
