Noisy sensor on a prototyping board

You need to step back and deal with this noise properly. Trying to cover it up in the firmware is the last resort. Something is making a mess somewhere.

Start by turning off everything except the receiver module and look at its output with a scope. Make sure the scope probe is properly grounded right at the receiver module ground. Make sure the logic analyser, the arduino, etc, are all off. Power the receiver from batteries if you have to. Is the noise now present on the receiver output? If so, then all the stuff that's off isn't at fault. Shut off as much other equipment in the vicinity as you can manage. Put a antenna tuned to your frequency (you only said ISM band, which doesn't tell us the frequency) on the receiver. Make a 1/2 wavelength dipole from wire if you need to. Keep poking around to find the source of the noise, which in this case is either bad module hookup or actual RF interference in your band. If the latter, there is little you can do about it.

If the signal is clean with just the module, then slowly add back other parts of the circuit one by one. Each time watch the output of the receiver on the scope. At some point you'll get the noise back, but now you'll know the source.

The best way to deal with any noise is to eliminate it at the source. Once noise gets added to your signal, there are likely parts of the noise that can't be distinguished from valid signal anymore. No amount of post processing can get rid of such noise.

I've worked on a project to implement the Kalman filter on an embedded system that was similar in hardware to the iNemo unit from STMicroelectronics.

Even if you can find these IMU (Inertial Measuring Unit) with 90% chance you will have to implement your algorithm by yourself; or if you're lucky, you can find someone that has the code. The problem is that this filter requires a lot of computation, and in our best experiment (using fixed point variables and trying to optimize the code) we were able to run it 45 times per second, in a STM32 at 72 MHz.

So maybe there is one, but as far as I know requires a good microcontroller or maybe a FPGA\ASIC.