This is not a reference, but a short proof.
We use the following (probably known, but see later) lemma on representing a symmetric tensor as a linear combination of rank-1 symmetric tensors.
Lemma. Let $A$ be a finite set, $K$ an infinite field. Denote by $\mathcal S$ the set of symmetric functions $p:A^n\to K$. Then $\mathcal S$ is the $K$-span of rank-one functions, that is, the functions of the type $h(x_1)h(x_2)\ldots h(x_n)$, where $h:A\to K$.
Proof. Note that the product of two rank-one functions is a rank-one function. Thus the linear space $\mathcal T$, generated by rank-one functions, coincides with the $K$-algebra generated by them.
We may suppose that $A\subset K$. For $k=0,1,\ldots,n$ denote $e_k(x_1,\ldots,x_n)$ the elementary symmetric polynomial, that is, $\varphi_t(x_1,\ldots,x_n):=\prod(1+tx_i)=\sum_{k=0}^n t^ke_k$. We identify $e_k$ and the corresponding element of $\mathcal S$. Choosing $n+1$ distinct values $t_1,\ldots,t_{n+1}\in K$ and solving the corresponding (Vandermonde's) linear system of equations we represent each $e_k$ as a linear combinations of $\varphi_{t_i}\in \mathcal T$. Thus $e_k\in \mathcal S$ for all $k=0,1,\ldots,n$. It is well known that $e_k$'s generate the algebra of symmetric polynomials (over any field). Thus any symmetric polynomial function belongs to $\mathcal T$. It remains to note that any symmetric function $f\in \mathcal S$ may be represented by a symmetric polynomial. Indeed, a symmetric function $f$ may be represented as $F(e_1,e_2,\ldots,e_n)$ for certain function function $F$ defined on the corresponding finite set (because the values of $e_1,\ldots,e_n$ determine the values of $x_1,\ldots,x_n$ up to permutation). $F$ in turn coincides with a polynomial function on this finite set. $\square$
Now we may prove your theorem for finitely supported function $i\mapsto p_i$. Due to Lemma it may be supposed to have the form $p_i=\prod_{k=1}^n H(i_k)$ for a certain finitely supported function $H$ on $\mathbb{N}$ (as OP, I denote here $\mathbb{N}=\{0,1,\ldots\}$). In this case both parts of your identity are equal to $\det (\sum_m H(m)A^m)$.
Comment. Lemma does not hold for finite fields. For example, if $A=K=\{0,1\}$. Then the function $x+y+z$ is not a linear combination of rank-one functions 1, $xyz$, $(x+1)(y+1)(z+1)$: if $x+y+z=a+bxyz+c(x+1)(y+1)(z+1)$, then for $y=0,z=1,x=a$ we get $0=1$. I must make a warning that in the subject-related paper "Symmetric tensors and symmetric tensor rank" by Pierre Comon, Gene Golub, Lek-Heng Lim, Bernard Mourrain (SIAM Journal on Matrix Analysis and Applications, 2008, 30 (3), pp.1254-1279) this statement, after equation (1.1), is stated for any field, although proved for complex numbers, and the proof uses that a non-zero polynomial has non-zero values.
In any case, you may always enlarge the ground field and safely think that it is infinite.
Proposition 1* does follow from that $\mathcal D_c$ sheafifies. For if $k(X) \cong k(Y)$, then $X$ and $Y$ are birational, so there are affine open subsets $U \subseteq X$, $V \subseteq Y$ such that $U \cong V$; then $\mathrm{Frac}(\mathcal D_c(X))$ is a localization of $\mathcal D_c(U) \cong \mathcal D_c(V)$, and similarly for $\mathrm{Frac}(\mathcal D_c(Y))$.
Theorem 1* follows from the following standard claim:
Theorem: if $X/k$ is an affine algebraic variety and a finite group $G$ acts freely on $X$, then the quotient map $\pi: X \to X/G$ is étale.
See e.g. Mumford, Abelian Varieties, §II.7 for a proof of this statement.
Since $\pi: X \to X/G$ is étale, it follows that $D\pi: T_X \to \pi^* T_{X/G}$ is an isomorphism.
Hence also $Sym_{\mathcal O_X} T_X \cong \pi^* Sym_{\mathcal O_{X/G}} T_{X/G}$. Now use the PBW theorem and $\Gamma(X,\mathcal O_X)^G = \Gamma(X/G, \mathcal O_{X/G})$ to conclude.
Best Answer
I guess you don't need approximate identity assumption to prove it (in any case what does approximate identity mean for a TRO?). I am not sure which paper of Kirchberg you are referring to, but indeed Kirchberg has proved that the sum of a closed left ideal and a closed right ideal is closed (for $C^*$-algebras) and this fact generalizes to TROs ($C^*$-spaces in Kirchberg's terminology) via the linking algebra construction. For the proof, see Section 4 of Kirchberg's paper "On restricted perturbations..." JFA 1995 (https://mathscinet.ams.org/mathscinet-getitem?mr=1322640).
Added: Associated with a closed left TRO ideal $I$ is a closed left ideal $$L:=\left[\begin{matrix} [VI^*] & I \\ I^* & [V^*I]\end{matrix}\right]$$ of the linking $C^*$-algebra $$A:=\left[\begin{matrix} [VV^*] & V \\ V^* & [V^*V]\end{matrix}\right].$$ Likewise for a right TRO ideal.