This paper extends the recently developed quasi-spectral model predictive static programming (QS-MPSP) to include state and control path-constraints and yet retain its computational efficiency. This is achieved by (i) formulating the entire problem in the control variables alone by (a) converting the system dynamics to an equivalent algebraic constraint and (b) converting the state constraints to equivalent control constraints, both of which is done by manipulating the system dynamics, (ii) representing the control variables in Quasi-spectral form, which makes the number of free-variables independent of time-grids and (iii) using a computationally efficient optimization algorithm to solve this low-dimensional problem. This generic computationally efficient technique is utilized next as an effective lead angle, and lateral acceleration constrained optimal missile guidance to intercept incoming high-speed ballistic targets with high precision successfully. Both of these constraints, as well as near-zero miss-distance, are of high practical significance for this challenging problem. Extensive three-dimensional simulation studies show the effectiveness of the newly proposed constrained QS-MPSP guidance algorithm. Six degrees-of-freedom simulation studies have also been carried out using autopilot in the loop to validate the results more realistically.