Can a BN CSM make a specific request for a soldier to get him PCS orders into his unit?

In theory yes... in practice it depends.
There has to be no shortages in your current unit, your grade and MOS.
There need to be a valid line number for the new unit, your mos and grade.
The Two CSM need to agree
The two CSM need to be wiling to make the request to HRC on your behalf..or at least gaining ask and loosing concur.
Neither of the two CSM's needs to have burned a bridge with HRC in the past.
And ...... up until the day you get to the new unit, newer, different, brighter minds at HRC can always come along and change the plan...
Lastly, once your at the new unit you "forced" HRC to put you in, new orders at the 18 month mark for PCS at 24 months is always a possibility for a mandatory fill position, DS, Recruiter, AIT PSG (or even more need as DS if they do away with AIT PSG as they are discussing now) or ROTC

This is true, but both sides and unit strengths have to support the request. Thank you for your service.

My best friend and roommate in korea talked to his CSM and 1SG for my upcoming PCS as my time was short since he left korea firsr. They agreed and i got orders to that same duty station. We then deployed together again he is also my daughters godfather. I was always thankful of them for this.