Another Mom who has faced this many, many times. In my son's case due to drugs. But none of mine have Traumatic Brain Injury (TBI).
I know little about Traumatic Brain Injury (TBI) but I believe that it is different for every person. Certainly I agree with the others that only you know your child and only you can make this decision. My choice was always to leave mine there to allow, as Nomad said, logical consequences. (In addition, to keep him off drugs.) Certainly he'll detox the hard way which I'm told is difficult. Again, part of logical consequences. But with your son's Traumatic Brain Injury (TBI) is he going to learn from this? Will he make a deal with you and keep it. Something like if you bail him out he'll go to inpatient rehab for a month? Mine would never keep a deal so I never bothered.
Bottom line is what are the pros and cons of leaving him there? My many years of experience with mine in jails and prisons is that the chance of him getting hurt in jail is minimal. That applies only if he is the type to not provoke people. Provoking other prisoners or guards can get him into trouble. In that case, a mental health facility is the only option. In terms of medications, if he's on any, his chance of reliably getting those in jail is variable.
I can say, in my experience, a lot of jails are helpful if you call them. Particularly if you explain your son's diagnosis. Some will refuse to discuss him specifically as he's an adult. But they'll answer general questions about what services they have (Dr on call?), how to get prescriptions to them, etc.
I hope you find peace with your decision.