Timo,
Time-Lenght-Value (TLV) extension carries Power via Media-Dependent-Interface (MDI). See snapshot from my lab
If the power budged is too low AP should not boot correctly as PSE allocated value won't fit classification signature at AP side.
Disabling the LLDP on each side will bring no negotiation, hence full AF power.
Which seems to be provided anyway.
Michale could capture LLDP on switch side to see what is AP asking for, but I am afraid it won't bring much more light. If other switches / ports and cables were tested the fault seems to be at AP side.
Regards,
Ondrej