A distribution is a continuous linear functional on the space $\mathcal{D}$ of test functions. A distribution $T$ is said to be translationally invariant if
$\left<T(x-a),f \right>:=\left<T,f(x+a) \right>=\left<T,f \right>$.
So is the property of translational invariance a property of the distribution $T$, or is it a property of the space of test functions $\mathcal{D}$? It seems to me that I could in principle make any distribution have translational invariance by suitably choosing the space $\mathcal{D}$ of test functions.
For example the distribution generated by the locally integrable $e^{i\omega x}$ is translationally invariant since
$ \left<e^{i\omega (x\pm 2\pi n)},f \right>=\left< e^{i\omega x},f \right>, $
but this is because $e^{i\omega(x\pm2\pi n)}=e^{i\omega x}$, and it has nothing to do with the properties of test functions in $\mathcal{D}$. It seems to me I could make also the distribution generated by $e^{-x^2}$ translationally invariant, by restricting $\mathcal{D}$ to translationally invariant smooth test functions satisfying $f(x+a)=f(x)$, namely
$\left<e^{-(x-a)^2},f\right>=\left<e^{-x^2}, f(x+a) \right>=\left<e^{-x^2},f \right>$.
So is also $e^{-x^2}$ translationally invariant?
Test functions over $\Bbb R^d $ are either compactly supported (for distributions) or belong to the Schwartz space (for tempered distributions; among other things, they are rapidly decaying on infinity), therefore, test functions can not be translationally invariant. In your terms, this would imply that "translational invariance" is a property of a distribution itself, not of the space of test functions.
You might want to define another space of test functions, say $C^\infty$ with some translational invariance (periodicity, for example), impose a topology, and then define continuous linear functionals (i.e. distributions) over that space. You will face some problems, of course (how to define a constant distribution, for starters), but this will at the end give you another space of "some-epithet distributions", where translational invariance is a property of the space of test functions.