How can I help a fellow soldier who wants to go to behavioral health, but is worried it will affect his career progression?

What will effect his career progression is if he does nothing; has a negative interaction with Law Enforcement or the Chain Of Command; and then has it all oooze out the sides.

Link him up with your unit MFLC the have a absolutely zero paper policy, unless there is a danger of them hurting themselves or someone else.

Have him go talk to them. The stigma of it hurting his career is just that. The Service wants HEALTHY people. That's what BH is doing, getting him healthy. Far better from a PREVENTIVE standpoint than a REACTIVE or CATASTROPHIC standpoint.