My soldier was set to PCS in a week but failed APFT and ABCP, our COC is barring him to keep him from PCSing. Can they do that?

Yes they can. Absolutely.
Once the bar is approved the AEA code "C" is applied to stabilize the SM and prevent movement.

SGT (Join to see) Both of the issues APFT and ABCP require a flag, but both flags are transferable>

B. APFT Failure: Commanders must initiate a Flag when a Soldier fails an APFT or
does not take the APFT within the time prescribed by existing regulation.
C. Noncompliance with Army Weight Control Program: Commanders must initiate this
flag when Soldiers do not comply with the Army Weight Control Program.

If the COC is considering other actions like separations, they can stop the PCS, if its JUST the APFT and ABCP flags then the SM will normally PCS. There is wiggle room for the COC as well. Reality is, things not aligned with a regulation can happen in the absence of anyone forcing the issue so even if the flag is transferable, the SM could have a PCS delayed.

I would be fighting hard to retain that SM, sending a problem to a gaining unit is both unprofessional and not in support of the Army's mission.
Chapter 2
Policy and Management of Flags
https://armypubs.army.mil/epubs/DR_pubs/DR_a/pdf/web/ARN10794_r600_8_2_Admin_Final.pdf

Maybe, but it would be a case of commander discretion rather than regulatory requirement, and the Soldier might have an IG case.
If this was a first failure of both and the Soldier was not previously flagged for a failure of at least one of these, APFT failure and noncompliance with ABCP are both transferable flags, (can transfer to a new unit and PCS).

For an APFT fail, you cannot initiate a bar or discharge until after the second record APFT fail. The paperwork to initiate a bar or discharge would require proof of two fails. Or if the Soldier was previously flagged for APFT in the last 36 months, then one APFT fail can result in bar or discharge.

For an ABCP flag, the Soldier must fail to make satisfactory progress and complete a medical evaluation before a bar to reenlistment is imposed (AR 600-9, 3-12). The paperwork to initiate a bar or discharge would require several 5500s showing lack of progress and the notes from the medical appt.
Or if the Soldier was previously flagged for ABCP fail in the last 36 months, a bar or discharge can be initiated after 90 days of non-compliance and a medical assessment.