Here is the instruction from the General Instruction of the Roman Missal:
In the dioceses of the United States of America, the following may also be sung in place of the psalm assigned in the Lectionary for Mass: either the proper or seasonal antiphon and psalm from the Lectionary, as found either in the Roman Gradual or Simple Gradual or in another musical setting; or an antiphon and psalm from another collection of the psalms and antiphons, including psalms arranged in metrical form, providing that they have been approved by the United States Conference of Catholic Bishops or the diocesan bishop. Songs or hymns may not be used in place of the responsorial psalm. (GIRM 61, emphasis added)
Show this instruction from the GIRM to the liturgy coordinator and the pastor. Explain your distress and show them that there is plenty of appropriate music available.
If that has no effect, then you will need to make a prudential judgment, weighing the relative merits of the witness of refusing to sing over a minor liturgical abuse versus the good influence you might have by staying. If you decide to stay, perhaps you should consider writing to the bishop to explain both the liturgical abuse and the apparent unwillingness of your pastor to respect your conscience.