Where to find a frequency doubler that works on signals between 5 and 500 KHz?
You can easily do it with your FPGA, no extra hardware is needed.

If you already have a working FPGA method for a fixed DC offset, you can allow the DC offset to be variable by using something called a "leaking integrator": From each sample s subtract a variable offset v. Modify the value of v for each sample by increasing v by one when s is positive and decreading v by one when s is negative. That way the value of v follows the changing DC offset and what remains after the subtraction is the AC component of the signal. Then you can square that signal.

[Edit] Lol, it's been so many years since I was in the university, I had to reassure myself that square of sine is in fact twice the frequency... which of course it is. :)