As others have noted, if you want DC and AC response you need a probe that uses a hall sensor and a coil of some kind. I personally use an old Tek P6042 current probe, DC-50 MHz, but it's optimized for clamping on small wires for troubleshooting SMPS (and you need several of them around for parts robbery). The probe head contains both a hall effect sensor and a tiny split core transformer. I am not sure this is necessarily the right tool for your particular use case.
The P6042 is a work of art more than a usable instrument. One bad drop and the transformer cracks and you no longer have a P6042. I had a perfectly working P6042 and one day a slight drop of the probe and it stopped working.
I luckily managed to scrounge a NOS replacement transformer sealed with the compensation resistors but that will never happen again.
And in any case the cable will break at the strain relief sooner or later.
It's fun to have and if you can use yours and not break it, fine, but it's like having a Fabergé egg... For display only.