You may be able to remove the decoupling caps on the FPGA side. But the noise is likely coming from the switching supply itself - these boards normally have pretty beefy switching supplies!
On the CW305 we add a bunch of filtering in-front of the shunt resistor (which does’t make sense for the “normal” use-case, so it’s unlikely to have enough filtering on the SP701):
Removing the decoupling caps on the FPGA itself should give you an idea how strong the signal could look. Right now you are probably seeing almost fully just switching power supply noise.
However - you may find a magnetic field probe works better with the FPGA. I can’t guarantee it would remove the noise problem, but it naturally cuts out some low-frequency noise (switch-mode noise), and is less sensitive to just the voltage noise itself.
Hope this helps!