The first thing I would try to do is figure out how "quod" works gramatically.
Quod erat demonstratum.
Thus this is demonstrated.
Definitely doesn't work. Find a Latin phrase that has an analogous sense.
Ad astra per aspera.
To the stars through desire.
(which Heinlein, in one story, famously corrected to
ad astra per ardua: to the stars through hard work)
Ah. So perhaps we want - if the two -a suffixes mean both those words are in the same case -
Ad liberta per astra
or something like that. Now, find out which declensions "astra" and "libertas" are in, and what cases "Ad" and "per" actually require. This would require an actual Latin grammar. Fortunately, Google books will help there, since people have been learning Latin in schools for a long time, and thus many of them are in the public domain.
Libertas, f. 3
Aster, m. 2
Apparently it would be
Ad libertas per astra
where both words are in the nominative case, as best I can tell from consulting one dictionary.
A Google search allowed me to find the motto of the West Park Secondary School: Libertas per diligentiam; Freedom through effort.
So "Libertas per astra" seems like it would do. Or "Libertas per stellarum", given these other possibilities:
Libertas per cultum
Freedom through education
Idler Academy motto
Libertas per scientiam
Freedom through knowledge
Benjamin Franklin American University
Libertas per veritatem
Freedom through truth
Lehigh University
From
here, there does seem to be a good reason for preferring "stellarum" to "astra"; "astra" is used 'poetically, or in more elevated prose', which makes sense when it is the goal, but not when it is the means.