It is possible but not guaranteed at a parish jail. In Louisiana, parishes function the same way counties do in other states, and Bossier Parish Jail is a local facility. County and parish-level jails are significantly less likely to permit inmate marriages than state or federal prisons, which typically have formal procedures in place for long-term inmates.
The first step is to call the facility and ask to speak with the chaplain. The chaplain is the person most likely to know whether marriages are permitted, what the process looks like if they are, and who needs to be involved to get it done. If the facility does allow it, expect to need a marriage license from the Bossier Parish Clerk of Court, a licensed officiant, and the warden's approval. There may also be requirements around who can be present as witnesses.
If Bossier Parish Jail does not permit marriages, your options are to wait until he is transferred to a state facility if that happens, or to consult with a local attorney about any other legal avenues.
Costs will depend on the marriage license fee set by the parish and any fees the officiant charges. The license fee in Louisiana is modest. Call the chaplain first and work from there.