Hi
As you have correctly pointed out, section 2.8 of the CIS340 excludes payments for construction operations by the governing body or head teacher on behalf of the LEA, which is also confirmed at The Scheme: contractors: schools (CISR12100) . You have asked whether the exemption still applies where the LEA themselves pay the invoices. The simple answer is yes, as a governing body are acting on behalf of the LEA, they could take the form of a committee or trust board and are essentially representing the LEA.
Hope this clarifies matters.