The most sensible way to approach this would be for there to be a neutral third party (a judge or perhaps a committee or board) that could decide when a district attorney's office should be compelled to pay for such testing.
The choice of whether to test cannot be left to the defendant. If it were, convicted criminals could flood the system with frivolous requests for DNA tests. This would be hugely expensive and would not yield any improvement in the justice of the system. In fact, it might cause those with real needs for DNA tests to have to wait in line while the frivolous cases are done.
The choice cannot be left to district attorneys either. District attorneys are typically deeply suspicious of claims of innocence since they have heard them all before. They are worried about their budgets and they are not inclined to believe tests are necessary. There is also something ethically wrong about allowing the prosecution to decide what sorts of chances the defense will get to prove its innocence.
Therefore, the only logical solution is to have a neutral third party decide on claims made by defendants.