Let me summarise the implications of [Blecher–Kaad–Mesland 2018, Section 3.4]. I'll assume that $A$ and $B$ are unital $C^\ast$-algebras.
Let $\mathcal{A} := \{a \in A \mid [D,a] \in B(H)\}$ endowed the so-called Lipschitz norm $\|a\|_D := \|a\| + \|[D,a]\|$. Then the inclusion $\mathcal{A} \hookrightarrow A$ is contractive with dense range closed under the holomorphic functional calculus (see [Mesland 2012, Section 4.2]). For convenience, let $\Omega_D$ denote the unital $C^\ast$-subalgebra generated by $A$ and $[D,\mathcal{A}]$, so that the inclusion $A \hookrightarrow \Omega_D$ makes $\Omega_D$ into a Hilbert $(A,\Omega_D)$-correspondence.
Let $\nabla : \mathcal{E} \to E \hat\otimes_A \Omega_D$ be a Hermitian connection, i.e., a $\mathbb{C}$-linear map with domain a dense subspace $\mathcal{E}$ of $E$ satisfying $\mathcal{E} \cdot \mathcal{A} \subset \mathcal{E}$ and $(\mathcal{E},\mathcal{E})_A \subset \mathcal{A}$, such that $$\forall e \in \mathcal{E}, \, \forall a \in \mathcal{A}, \quad \nabla(ea) = \nabla(e)a + e \hat\otimes [D,a],\\ \forall e_1,e_2 \in \mathcal{E}, \quad (e_1 \hat\otimes 1,\nabla(e_2))_{\Omega_D} - (\nabla(e_1),e_2 \hat\otimes 1)_{\Omega_D} = [D,(e_1,e_2)_A].$$ Then $\nabla$ is, in fact, a closeable operator, and the domain $\mathcal{E}_\nabla$ of the closure satisfies $\mathcal{E}_\nabla \cdot \mathcal{A} \subset \mathcal{E}_\nabla$ and $(\mathcal{E}_\nabla,\mathcal{E}_\nabla)_A \subset \mathcal{A}$.
Now, for any $b \in B$, such that $b \cdot \mathcal{E}_\nabla \subset \mathcal{E}_\nabla$, we can define $\delta_\nabla(b) : \mathcal{E}_\nabla \otimes^{\mathrm{alg}}_\mathcal{A} \Omega_D \to E \hat\otimes_A \Omega_D$ by $$\forall e \in \mathcal{E}_\nabla, \, \forall \omega \in \Omega_D, \quad \delta_\nabla(b)(e \hat\otimes \omega) := \nabla(be)\omega - (b \hat\otimes 1)\nabla(e)\omega.$$ Now, let $$\mathcal{B} := \{b \in B \mid b \cdot \mathcal{E}_\nabla + b^\ast \cdot \mathcal{E}_\nabla \subset \mathcal{E}_\nabla, \; \delta_\nabla(b),\delta_\nabla(b^\ast) \in B(E \hat\otimes_A \Omega_D)\}, $$ and observe that $\mathcal{B}$ is dense in $B$ since it contains the ket-bra $\lvert e_1 \rangle \langle e_2 \rvert$ for every $e_1,e_2 \in \mathcal{E}_\nabla$. It turns out that $\delta_\nabla : \mathcal{B} \to \operatorname{End}_{\Omega_D}(E \hat\otimes_A \Omega_D)$ is a closed $\ast$-derivation densely defined on $B$; indeed, if we endow $\mathcal{B}$ with the Lipschitz norm $\|b\|_{\nabla} := \|b\| + \|\delta_\nabla(b)\|$, then the inclusion $\mathcal{B} \hookrightarrow B$ is contractive with dense range closed under the holomorphic functional calculus.
At last, define $1 \hat\otimes_\nabla D : \mathcal{E}_\nabla \otimes^{\mathrm{alg}}_{\mathcal{A}} \operatorname{Dom} D \to E \hat\otimes_A H$ by $$ \forall e \in \mathcal{E}_\nabla, \, \forall h \in \operatorname{Dom}(D), \quad 1 \hat\otimes_\nabla D(e \hat\otimes h) := \nabla(e)h + e \hat\otimes Dh; $$ observe, then, that for any $b \in \mathcal{B}$, $$ \forall e \in \mathcal{E}_\nabla, \, \forall h \in \operatorname{Dom}(D), \quad [1 \hat\otimes_\nabla D,b \hat\otimes 1](be \hat\otimes h) = \delta_\nabla(b)(e \hat\otimes 1)h. $$ The methods of [Brain–Mesland–Van Suijlekom 2016] now suffice to show, in particular, that $1 \hat\otimes_\nabla D$ is essentially self-adjoint, yielding a spectral triple $(B,E \hat\otimes_A H,1 \hat\otimes_\nabla D)$ with $\mathcal{B} = \{b \in B \mid [1 \hat\otimes_\nabla D,b] \in B(E \hat\otimes_A H)\}$, precisely since $[1 \hat\otimes_\nabla D,\cdot]$ is essentially the same thing as $\delta_\nabla$.