Have you looked into
carry cots? This picture is just one example.
What you would do is still try to request a bulkhead seat and lay the cot on the floor.
The only downside is that they are somewhat bulky, but if you're travelling with another adult, it would be manageable. They would store pretty easily in the overhead bin for takeoff/landing.
$70 here (Oops, this is an Australian site.) Search for a Bertini reseller in the US, and they may be able to get you one like this.