What is the condition that a test function $f(t^{\prime},t^{\prime\prime})$ has to satisfy in order that:
$$\lim_{t\rightarrow t_{0}}\int_{t_{0}}^{t}dt^{\prime}\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\:f(t^{\prime},t^{\prime\prime})\,\delta^{\prime}(t^{\prime\prime}-t^{\prime})\,=\,0\quad?$$ where
$$\delta^{\prime}(t^{\prime\prime}-t^{\prime})=\frac{d}{dt^{\prime\prime}}\delta(t^{\prime\prime}-t^{\prime})$$ with $\delta(t^{\prime\prime}-t^{\prime})$ is the Dirac delta function.
(Partial answer)
The main "problematic" step with this expression will occur when doing the integration by parts. Indeed, we will get : $$ \left.\int_{t_0}^{t'} f(t',t'')\delta'(t''-t') \,\mathrm{d}t'' = f(t',t'')\delta(t''-t') \right|_{t''=t_0}^{t''=t'} - \int_{t_0}^{t'} f_y(t',t'')\delta(t''-t') \,\mathrm{d}t'', $$ where $f_y$ denotes the derivative of $f$ with respect to its second argument. The first boundary term, namely $f(t',t')\delta(0)$, will be divergent unless you impose $f(t',t') = 0$. I let you finish the derivation, but it seems to me that no more constraint is needed.