Your question is entirely covered by Section 2 of [Brain–Mesland–Van Suijlekom][1], but the fgp case is simple enough to ultimately boil down to folklore proved by [Chakraborty–Mathai][2]. Let me summarise what happens, while incorporating some technical simplifications from [Blecher–Kaad–Mesland][3]. For convenience, let $\Omega_D$ be the unital $C^\ast$-subalgebra of $B(H)$ generated by $A$ and $[D,\mathcal{A}]$.

 1. Let $\mathcal{A} := \{a \in A \mid a \operatorname{Dom}(D) \subset \operatorname{Dom}(D),\, [D,a] \in B(H)\}$ be given the Lipschitz norm $\|a\| := \|a\| + \|[D,a]\|$. Then $\mathcal{A}$ defines an involutive operator algebra and the inclusion $\mathcal{A} \hookrightarrow A$ is completely bounded with dense range that is closed under the holomorphic functional calculus; in particular, it follows that $M_N(\mathcal{A})$ is dense in $M_N(A)$ and closed under the holomorphic functional calculus for any $N \in \mathbb{N}$. You should think of $\mathcal{A}$ as defining a Lipschitz structure on the NC space $A$.
 2. Let $\mathcal{E}$ be a dense subspace of $E$ satisfying $\mathcal{E} \cdot \mathcal{A} \subset \mathcal{E}$ and $(\mathcal{E},\mathcal{E})_A \subset \mathcal{A}$. By part 1 and the properties of $\mathcal{E}$, we can find $A$-module generators $\{\xi_1,\dotsc,\xi_N\} \subset \mathcal{E}$ for $E$, such that 
$$
 \forall e \in E, \quad e = \sum_{i=1}^N \xi_i \cdot (\xi_i,e)_A.
$$
Thus, if $p := \left((\xi_i,\xi_j)_A\right)_{i,j=1}^N \in M_N(\mathcal{A})$, then $e \mapsto \left((\xi_i,e)_A\right)_{i=1}^N$ defines an isomorphism $E \cong pA^N$ of Hilbert $A$-modules that restricts to an isomorphism $\mathcal{E} \cong p\mathcal{A}^N$ of pre-Hilbert $\mathcal{A}$-modules; this now makes $\mathcal{E}$ into a (finitely generated) projective operator module over $\mathcal{A}$ in a manner that depends on the choice of $\{\xi_1,\dotsc,\xi_N\}$ only up to completely bounded isomorphism. You should think of $\mathcal{E}$ as defining a Lipschitz structure on the NC vector bundle $E$; since $E$ is fgp, this Lipschitz structure is canonically induced by the choice of Lipschitz structure on $A$.
 3. Let $\mathcal{B} := \{b \in B \mid b \cdot \mathcal{E} \subset \mathcal{E}\}.$ Since
$$ 
 \mathcal{B} = \left\{b \in B \mid \left((\xi_i,T\xi_j)_A\right)_{i,j=1}^N \in M_N(\mathcal{A})\right\},
$$
it follows that $\mathcal{B}$ is $\ast$-closed and that the $\ast$-isomorphism $B \cong p M_N(A) p$ induced by the Hilbert $A$-module isomorphism $E \cong p A^N$ restricts to a $\ast$-isomorphism $\mathcal{B} \cong p M_N(\mathcal{A}) p$. Thus, $\mathcal{B}$ can be topologised as a closed $\ast$-subalgebra of the involutive operator algebra $M_N(\mathcal{A})$, thereby making $\mathcal{E}$ into a Lipschitz $(\mathcal{B},\mathcal{A})$-correspondence. In particular, you can view $\mathcal{B}$ as defining a Lipschitz structure on the NC space $B$ compatible with the Lipschitz structure $\mathcal{A}$ on the NC space $A$.
 4. Let $\nabla : \mathcal{E} \to E \hat\otimes_A \Omega_D$ be a *Hermitian connection*, i.e., a $\mathbb{C}$-linear map satisfying 
$$
 \forall e \in \mathcal{E},\, \forall a \in \mathcal{A}, \quad \nabla(ea) = \nabla(e)a + e \hat\otimes [D,a],\\
 \forall e_1,e_2 \in \mathcal{E}, \quad (\nabla(e_1),e_2 \hat\otimes 1)_{\Omega_D} +(e_1 \hat\otimes 1,\nabla(e_2))_{\Omega_D} = [D,(e_1,e_2)_A];
$$
for example, the *Graßmann connection* $\nabla_0$ induced by the frame $\{\xi_1,\dotsc,\xi_N\}$ is defined by
$$
 \forall e \in \mathcal{E}, \quad \nabla_0(e) := \sum_{i=1}^N \xi_i \hat\otimes [D,(\xi_i,e)_A],
$$
and indeed, if $\nabla$ is any other Hermitian connection, then $\nabla = \nabla_0 + \omega$ for $\omega \in B(E,E \hat\otimes_A \Omega_D)$ defined by
$$
 \forall e \in \mathcal{E}, \quad \omega(e) := \sum_{i=1}^N \nabla(\xi_i)(\xi_i,e)_A.
$$

 5. Given a Hermitian connection $\nabla$, define $1 \hat\otimes_\nabla D : \mathcal{E} \otimes^{\mathrm{alg}}_{\mathcal{A}} \operatorname{Dom}(D) \to E \hat\otimes_A H$ by
$$
 \forall e \in \mathcal{E}, \, \forall h \in H, \quad 1 \hat\otimes_\nabla D(e \hat\otimes h) := \nabla(e)h + e \hat\otimes Dh = \omega(e)h + 1 \hat\otimes_{\nabla_0}D(e \hat\otimes h),
$$
so that by Section 2 of Chakraborty–Mathai, the operator $1 \hat\otimes_\nabla D$ is essentially self-adjoint and defines a spectral triple $(B,E \hat\otimes_A H,1 \hat\otimes_\nabla D)$ with Lipschitz algebra $\mathcal{B}$. In particular, $1 \hat\otimes_\nabla D$ is a bounded perturbation of $1 \hat\otimes_{\nabla_0} D$, where, by a direct computation,
$$
 \forall b \in \mathcal{B}, \, \forall e \in E, \, \forall h \in H, \quad [1 \hat\otimes_{\nabla_0} D,b](e \hat\otimes h) = \sum_{i,j=1}^N \xi_i \hat\otimes [D,(\xi_i,b\xi_j)_A](\xi_j,e)_A h.
$$

To conclude, the missing ingredient was a Lipschitz structure on the fgp $A$-module $E$, i.e., a dense subspace $\mathcal{E}$ of $E$, such that $\mathcal{E} \cdot \mathcal{A} \subset \mathcal{E}$ and $(\mathcal{E},\mathcal{E})_A \subset \mathcal{A}$. Because $E$ is fgp, this suffices to pick out a dense $\ast$-subalgebra $\mathcal{B}$ of Lipschitz elements of $B$ and to pin down any Hermitian connection $\nabla$ on $E$ with domain $\mathcal{E}$ up to bounded perturbation, so that $\mathcal{B}$ will necessarily have bounded commutators with $1 \hat\otimes_\nabla D$; indeed, by using a normalised tight frame for $E$ consisting of vectors in $\mathcal{E}$, you can boil everything down to the compatibility of $\mathcal{A}$ with $D$.

  [1]: https://arxiv.org/abs/1306.1951
  [2]: https://arxiv.org/abs/0804.3232
  [3]: https://arxiv.org/abs/1703.10063