As UllaH pointed out, the longname for the BSSP585 compset is
SSP585_CAM60_CLM50%BGC-CROP_CICE_POP2%ECO%ABIO-DIC_MOSART_CISM2%NOEVOLVE_WW3_BGC%BDRD
The BDRD at the end of this longname describes how CO2 is used in the experiment.
The BD indicates that the biogeochemistry modules in the surface components use diagnostic (i.e. prescribed) CO2.
The RD indicates that atmospheric radiation computations use diagnostic (i.e. prescribed) CO2.
If you replace BDRD in the compset longname with BPRP, then the biogeochemistry modules in the surface components and the atmospheric radiation computations will both use prognostic (i.e. computed) CO2. It sounds like this is what you want.
I've attached some slides from the BGC practical session of the CESM tutorial that (somewhat tersely) explains this. In particular, please see slide 3.
Note that I advise against initializing a BPRP experiment from a BDRD experiment. The atmospheric CO2 tracers in a BDRD compset are usually out of balance with the carbon cycle. So you would get large transients in carbon fluxes at the beginning of your experiment. You should initialize using a reference case that was BPRP. The inputdata repository has restart files for 2015-01-01 from b.e21.BHIST_BPRP.f09_g17.CMIP6-esm-hist.002 that suffices for this.
At one point we had submitted BPRP SSP585 results from CESM2.1 to CMIP6. However, a bug was found in the forcing and the results were retracted. The experiment has not yet been rerun.
I think the absence of this compset in the CESM 2.1 release series is just an oversight. I don't recall having a reason to intentionally omit this compset.
