Is it possible to get my SPC laterally promoted to CPL now that SGT does not have to complete BLC before getting promoted?

No. Army policy hasn't changed (for RA and USAR) since the implementation in 2022 (AD 2021-17*, para 5c).

• In order to attend BLC, a SPC must be recommended for promotion.
• When the Solider graduates BLC, then they will be laterally appointed to CPL

As the Army has given prescriptive guidance, there aren't "alternative methods" that can be used to get around it (again, for RA and USAR Soldiers) - "The CPL rank will be worn only by Soldiers who are both recommended for promotion by a unit promotion board and graduates of the BLC, regardless of the grade of position to which they are assigned" (AD 2021-17, 5b(1)).
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* Army Directive 2021_17 _ https://armypubs.army.mil/epubs/DR_pubs/DR_a/ARN32643-ARMY_DIR_2021-17-000-WEB-1.pdf

A Lateral Promotion, specific to the unit, is merely a 4187 approved by the unit commander.

During a period of time they made it mandatory to serve as CPL for a certain amount of time ?
But it seems promotions are different now where SPC go straight to SGT rank most likely because of promotion points.
Your squad leader should inform about BLC get with your local chain of command