Consider a sequence of functions $(q_{\eta})_{\eta\in\mathbb{R}_{>0}}\subseteq L^{1}(\mathbb{R};\mathbb{R}_{\geq0})$ so that there exists $q^{*}\in L^{1}(\mathbb{R};\mathbb{R}_{\geq0}):\ q_{\eta}\rightarrow q^{*} \text{ in } L^{1}(\mathbb{R})$ for $\eta\rightarrow 0$. Assume in addition that there exists $C\in\mathbb{R}_{>0}:\ \sup_{\eta\in\mathbb{R}}|q_{\eta}|_{TV(\mathbb{R})}\leq C$. Then, we want to show the following:
For $\phi\in C^{1}_{\text{c}}(\mathbb{R})$ assume that the sequence \begin{align} \tfrac{1}{2\eta} \int_{\mathbb{R}} q_{\eta}(x)\Big(q_{\eta}(x+\eta)-q_{\eta}(x-\eta)\Big)\phi(x)\Big(q_{\eta}(x+\eta)+q_{\eta}(x)+q_{\eta}(x-\eta)\Big)\mathrm{d} x \tag{1}\label{eq:1} \end{align} converges for $\eta\rightarrow 0$ to $-\int_{\mathbb{R}} (q^{*}(x))^3\phi_{x}(x)\;\mathrm{d} x,$ i.e. $$ \lim_{\eta\rightarrow 0}\tfrac{1}{2\eta} \int_{\mathbb{R}} q_{\eta}(x)\Big(q_{\eta}(x+\eta)-q_{\eta}(x-\eta)\Big)\phi(x)\Big(q_{\eta}(x+\eta)+q_{\eta}(x)+q_{\eta}(x-\eta)\Big)\mathrm{d} x\\=-\int_{\mathbb{R}} (q^{*}(x))^{3}\phi'(x)\;\mathrm{d} x,\label{eq:2} \tag{2} $$ can we say anything about the convergence of \ref{eq:1} when leaving out one of the terms in the fourth factor, i.e. can we say anything about the convergence of$$ \lim_{\eta\rightarrow 0}\tfrac{1}{2\eta} \int_{\mathbb{R}} q_{\eta}(x)\Big(q_{\eta}(x+\eta)-q_{\eta}(x-\eta)\Big)\phi(x)\Big(q_{\eta}(x+\eta)+q_{\eta}(x-\eta)\Big)\mathrm{d} x=?? $$ Clearly, the right candidate for the convergence would presumably be (compare to \ref{eq:2}) $-\tfrac{2}{3} \int_{\mathbb{R}}(q^{*}(x))^{3}\phi'(x)\;\mathrm{d} x$ however, we fail to show this.
The problem can also be interpreted as a question about convergence of finite difference approximations of derivatives as the second factor in \ref{eq:1}, $\tfrac{q_{\eta}(x+\eta)-q_{\eta}(x-\eta)}{2\eta}$ can be seen as the approximation of $q_{\eta}'(x)$ (which might in our setting exist for $\eta\in\mathbb{R}_{>0}$ but which does NOT exist for $\eta\rightarrow 0$ or only in a measure sense).
EDIT: For a simplified version of this question see Finite difference approximations of serivatives of polynomials of BV functions