if you will be covering the turf with mulch, then it will die anyways, I would just mow real low.
Landscape fabric is a scam in most instances. It does not help with weed control, and keeps the fine roots out of the mulch, and worms and such from moving between the mulch and soil.
I would maybe use kraft paper, or newsprint, then a layer of compost then the dry chip mulch.
just putting chip over the turf will take up most of the Nitrates in the soil while decomposing. Using an initial layer of decompsed organics (Compost) will protect the shrubs root system from this.