Modern Code limits what that circuit could serve
If the installation is in the last 20 years or so, modern Code would apply, and that limits the possible uses of a bathroom receptacle circuit. It must pick one of these rules and follow it:
- the circuit may serve ONLY receptacles ONLY in bathrooms plural.
- The circuit may serve ONLY receptacles in 1 bathroom, but can serve hard-wired loads in that same bathroom also.
So that is what I would expect in a house modern enough to be sold with a GFCI breaker.
GFCIs care about difference in current
Current flows in loops. In a perfect world, hot current should exactly equal neutral current. GFCIs work very simply: they compare hot and neutral current, and determine the difference in current (which should be zero). >5mA and they trip.
You are measuring current on a hot wire or a neutral wire. One is meaningless without the other, at least as far as troubleshooting a GFCI. If you are measuring 30mA on the hot wire, and unbeknownst to you there is also 30mA on the neutral wire, that is working as intended. (Since the GFCI is holding, neutral current must be >25mA or <35mA, yes?)
What will trip a GFCI is >5mA current from hot to something else or neutral to something else. I would say "ground" but I don't want you to stick an ammeter on the ground wire. It may be transiting the water pipes to the system grounding electrode and back that way. Or current may be moving from this circuit's hot to another circuit's neutral, or vice versa, because a nitwit borrowed a neutral (a code violation even if GFCIs aren't involved).
What should test out
If you disconnect hot and neutral wires from the GFCI, (but leave the curly neutral pigtail on it), a couple of things should be true.
First, the GFCI breaker should hold. If the GFCI (any type) tripped with no "Load" wires on it, that's a defective GFCI. The GFCI holding is a very positive sign that it's not the $40 GFCI.
Second, if you gather hot-neutral together and connect them to each other, and measure that to Ground, there should be no conductance at all. That is to say, infinity ohms. You're lucky if a cheap DVM fails the circuit at this point, because that gives you something to test for. But many insulation failures behave like a VBO - insulate at <=X voltage but conduct above X voltage. To detect those, you need an insulation tester aka a "mega-ohm-meter" or "MEGGER" which is an ohmmeter with a very high test voltage of 250, 500 or 1000 volts depending on machine.
Fortunately, your panel already contains a MEGGER with a 120V test voltage and a 5mA detection threshold! Yeah. That GFCI.
In fact, if you wanted to gather the circuit's hot and neutral and put them both on the hot terminal of the GFCI, that would be a fairly effective insulation test! (it would put a higher test voltage on neutral than it normally gets). That won't hurt any of the equipment.
The next step, then, is to figure out the circuit's route and break it (both hot and neutral) near the midpoint. Test with only half the circuit connected. Does it still trip -> it's in the first half. If it holds -> it's in the second half. Divide the circuit at the 1/4 or 3/4 point (depending on the results of the first test) and test again. Continue to "divide and conquer" until you have identified the device or cable run at fault.