build the risers and stage first. There is no need to put wall treatments between the riser/stage and the walls just the 1/4 inch gap as you are planning.
There is also some concern whether to put the carpet in before you do the wall treatments. If your space is tight and you have some doubts about the skill level of the carpet layers (Like HD pot luck) it might be better to have the carpet in before you put the fabrics on the wall. Furring strips and Linacoustic OK but If you think the GOM or other fabric may be at risk then wait.
Now if you have carpet installed before furring strips and baseboard moulding be sure to have them install the tack strips the required distance from the wall so that the carpet doesn't get captured forever beneath your moulding and furring.