HeartPrinter is a novel under-constrained 3-cable parallel wire robot designed for minimally invasive epicardial interventions. The robot adheres to the beating heart using vacuum suction at its anchor points, with a central injector head that operates within the triangular workspace formed by the anchors, and is actuated by cables for multi-point direct gene therapy injections. Minimizing cable tensions can reduce forces on the heart at the anchor points while supporting rapid delivery of accurate injections and minimizing procedure time, risk of damage to the robot, and strain to the heart. However, cable tensions must be sufficient to hold the injector head's position as the heart moves and to prevent excessive cable slack. We pose a linear optimization problem to minimize the sum of cable tension magnitudes for HeartPrinter while ensuring the injector head is held in static equilibrium and the tensions are constrained within a feasible range. We use Karush-Kuhn-Tucker optimality conditions to derive conditional algebraic expressions for optimal cable tensions as a function of injector head position and workspace geometry, and we identify regions of injector head positions where particular combinations of cable tensions are optimally at minimum allowable tensions. The approach can rapidly solve for the minimum set of cable tensions for any robot workspace geometry and injector head position and determine whether an injection site is attainable.