I have a riser and bar just as you describe in my basement.
I kick myself routinely for not trenching the concrete for electrical connections to the wall. I did that in another location in my basement and I just dropped the ball and didn't do it. Trenching involves using a big hammer (Jack hammer) drill to break up the concrete. running the line then filling the trench back up with concrete. One day job. You need to rent the hammer, it runs on 120V.
I ended up using a rubber cord cover (Corduct) that lays on the surface of the carpet.
You can leave the carpet as is just don't expect to remove the riser in the future and reuse the carpet, it will be crushed
studs 16 inches OC use two or three layers of decking material (OSB or plywood) to keep it quiet and stuff with insulation, think about step lights for safety.