Rather than putting that into the constitution (and if you can't see how a right-wing Supreme Court would transform such a provision into corporate welfare, you're being a tad naive), the obvious solution would be setting up such an office via ordinary legislation.
Of course I see how it could be turned against itself, but ordinary legislation is hardly better; then you're making its existence contingent on the legislature continuing to support it, and Congress or the state legislature could easily change it into a form of corporate welfare anyway. It's a damned if you do, damned if you don't situation.
Or you could simply do what other jurisdictions do, and generally have the court force the losing side to pay the winner's court costs. That limits cases to ones people think they can actually win.
I'm not sure what your reference to "other jurisdictions" is supposed to mean, because that is common in the United States as well. The problems with this approach is that it does not operate to promote litigation in the public interest, and in fact in certain ways
suppresses litigation that might be in the public interest. First, obviously no one is paying anything until the case is resolved, so litigants with limited resources (that is, many of them in cases of special public interest) cannot even initiate litigation unless they find a lawyer willing to take them on contingency or a group which is litigating cases for some political reason of their own. And, again, this will not happen every time someone has a good case the litigation of which might be in the public interest, but will be shaped by the presence (or absence) of the lawyers or legal groups in question and their own agendas in supporting litigation.
Second, just because someone has a solid case and a good basis for expecting a victory doesn't mean that they
will win. Again, for a litigant with limited resources a loss is devastating because legal fees are expensive, especially if your opponent is a large firm or government with very large resources and numerous lawyers and other legal staff working for them. So litigants in this position will be strongly deterred from litigating unless someone else is covering the costs or they have such an incredibly strong case that victory is not only likely, but virtually certain. Otherwise, they're running a significant risk of being permanently bankrupted and destroying their life if they sue, again even if they have a good, solid case that has a reasonable chance of victory but not a certainty.
The only advantages to this arrangement are that it does operate to suppress frivolous litigation and in the case that litigation is pursued and victory is obtained by the less-resourced side it does help compensate them for their trouble. But both of these could be achieved through other means that might be less harmful to the pursuit of suits against the rich and powerful by the weak and powerless.