OP should not fixate on FRA as a connection point. Hand luggage allowances are ultimately limited by the carrier (and might even be stricter than what is allowed by airport authorities in some situations). Thus, check the carry-on allowance for each flight's carrier. That is what will apply for the flight in question.

Although there may be multiple checks, it is ultimately gate agents who see a passenger with their bag(s) who permit a passenger to board. Or not.

If you are someone who intends to chance violating the law or policy as the case may be, you also need a Plan B. What will you do if you show up overweight or over count at the gate and the gate agent will not permit you to board? Will you permit the agent to gate check your bag or will you choose not to board, possibly forfeiting the remaining segment(s) of your ticket?