Can a Soldier be removed from a temporary profile in order to attend BLC?

What does the doctor say about the injury? This should be a medical decision.

You cannot remove a Soldier from a profile or modify it. You and your leadership can recommend it, but the issuing profililers (i.e., the medical doctors with the authority to issue a profile) are the only ones that can issue, modify, or terminate it prior to its expiration.

Profiles are evaluations by medical personnel on what limits should be applied to servicemembers. 'In the old days', a commander had the authority to go against the profile recommendations of the medical community, but assumed risk by doing so. "Now days", that authority still exists, but is largely overridden by policy/regulations from higher commands stipulating that unit commanders will follow the restrictions on the profile in all but some limited situations.

Specific to PME courses - TRADOC is clear that a Soldier on a temporary profile, which was not caused by an operational deployment and prevents full participation in the course, will be removed or deferred from school attendance consideration by their immediate commander until they can complete all course graduation requirements (see BLC ISAP*).

That last part is the critical part. A temporary profile in itself will not prevent a Soldier from attending a TRADOC PME course unless it prevents them from meeting the course requirements. However, if the profile the Soldier has would not allow them to complete an ACFT using alternate events or preclude participation in any of the other required training, then it would bar them from attending.

If you feel the Soldier's profile is overly restrictive and those restrictions prevent them from meeting the graduation requirements, the only course of action you have is to ask your leadership to discuss modifying it with the profiling officials. If the profiling officials don't change the restrictions then the Commander is required to defer/remove them from the course.
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* BLC Student Guide (July 2023) - https://home.army.mil/campbell/8916/9506/3608/BLC_Student_Guide_01_July_23.pdf
* BLC Individual Student Assessment Plan (ISAP) - https://home.army.mil/liberty/application/files/4015/5475/8109/BLC_ISAP_March_2019.pd.pdf

Better safe than sorry.

The soldier needs to return to the medical treatment facility to discuss his condition with the doctor. They can discuss the sort of activities required in BLC and whether it is reasonable for him to perform those activities.

If the doctor decides he is medically fit to perform duty without limitations, the doctor will cancel the profile and return the soldier to full duty. The profile is entirely a medical decision.

Meanwhile you can discuss the situation with the soldier. Ensure that he understands that he would be expected and required to perform all the physical activities of BLC. There is no way for him to 'take it easy' or 'be careful' while attending BLC. If he exacerbates his condition he would be unable to complete BLC.

If this Soldier has a reservation, have your BN or BDE OPS AGRs find their folks with future reservations after this profile ends, and see if one wants to swap with this guy's earlier reservation.
If not, send up the cancel, and when it's canceled, book a new reservation.

Only by the issuing medical provider.

BLC isn't a smoke session, but they should still get the provider to assess the profile.

Hello, they are gonna get an MRI completed in a couple of weeks. This injury has been since March and has not gotten better even with PT.