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

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.

This sort of happened to me. Before I was given pinpoint orders at my first duty station, a friend of mine went to the BN CSM and asked on my behalf for me to be put in that BN rather than another. He agreed and it was a done deal.

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

You have been in the military, what just over 3 years? This is your first duty station and second CSM you have worked under? Go where the Army tells you experience life and maybe even better leadership than you have to date.

In general, HRC manages at the installation level though there are a few units that are managed separately. For example, at Fort Bragg there are three "accounts" or DMLs; BRG-Fort Bragg as a whole, SOC-Special Operations Command units on Fort Bragg, and there is an INSCOM account there as well. HRC can assign you to those DMLs and in a few cases to a sub-DML. Then it is up to the CSM to negotiate with the installation manpower managers to get someone they want. At Fort Huachuca years ago, this was referred to as "the meat market meeting". All the key CSMs and the installation manpower managers sat down once a month and worked out the distribution of all the incoming Soldiers. A lot of bartering took place there. I liked the S-1 Sergeant that I had so I traded an incoming S-1 Staff Sergeant for a Quartermaster Specialist (E-4) that I needed but wasn't authorized. SGM Marquez described the other aspects of the assignment process.

In 1978, CSM Gilbert Broner assured me that 1SG Fox. A Co, 2nd Bn, 'Buffalo', 17th Infantry, 7th ID,Ft Ord, was slotted for the SGM Academy. When I asked how he knew, Gil said, "I made a phone call". I am on the right of the photo, CSM Broner is next to me.