Well, the OP will be checking in with KL's agency staff at BHX so they wouldn't be dealing with EY check-in staff at all.
As this is an interline ticket sold by EY in full knowledge of - and in cooperation with - KLM, the check-in agent should be presented with the relevant EY baggage rules and the corresponding extra fees chargeable if and when luggage over and above the ticket allowance is presented.

My point earlier, which I probably did not manage to convey clearly, is that even if the KL check-in agent wanted to give the FB benefit of a free extra bag for the BHX-AMS sector, they would still be required to take payment as required by Etihad's fees. I genuinely don't think the agent would then be able to "recalculate" that fee to apply from AMS only, but even if they could, I imagine the same, or a broadly similar, fee would still apply; the fee is probably identical so there would realistically be no saving, even if getting BHX-AMS for "free"

I would expect the computer to spit out a number and I don't know what discretion, if any, the check in agent would have to waive, override or reduce this fee. They are also unlikely to incorrectly enter into the system that the money was collected if they themselves do not take payment, because then they* (either KLM or the airport handling comany) will be out by that amount which gets transferred to Etihad. Or the OP may be called to the desk at AMS or AUH if the EY agents suspect or see any irregularities with the luggage.

My advice to the OP would be to pack only what you need and to expect to have to pay more if you go over your EY allowance. Forget about your Flying Blue perks because they don't apply to Etihad-issued tickets even as SkyPriority benefits may indeed apply.