From the description, it sounds like the neutral may be shared between the two circuits, which you can't do with a GFCI. The GFCI is looking for an imbalance of current on the hot and neutral lines, so if the neutral is shared with another circuit, there will be some current returning on the neutral that did not go out on the hot.
If you can trace the circuit and find the connection between the neutrals on the two circuits, and isolate them, that would be the easiest and cheapest fix. If the neutral is shared via a multiwire branch circuit from the breaker and can't be split, you can change the gfci breaker to a standard breaker and install a gfci outlet after the neutral splits for the two branches. And I'm also seeing two pole gfci breakers which would allow you to replace the two breakers with one that provides gfci coverage to both circuits and should handle the shared neutral, which may be a good fall back option if you can't locate the shared neutral.
As well it should.
Normally, loads are connected between hot and neutral. Appliances are not supposed to connect hot or neutral to ground; ground is only a shield.
The GFCI compares current on the hot and neutral wires. They should be the same. If they are not, current has found another route, possibly through the grounding system (which isn't supposed to happen) and potentially through some poor human.
Circuit testers are trying to test whether ground is connected... cheaply. They mis-use "hot" as a power source, by connecting a light bulb between hot and ground. If ground is connected correctly, this will light.
In other words, it intentionally creates a hot-ground fault (by sticking a light bulb there). This is exactly the condition GFCIs are designed to detect.
I'm not talking about any ground-fault-test the tester may also have.
So why do testers often work? With a perfect GFCI, they wouldn't work. I suspect it is because GFCI's have detection thresholds above zero, and that is often enough for these testers to "get away with it". I even think there may be a tacit agreement among manufacturers for this, but obviously, lower sensitivity impinges safety. Remember a shock which only stuns you can kill you with secondary effects like falling or drowning.
So either your GFCI is pretty good, or your tester is pretty bad.
Best Answer
Look at the wiring on the connection block on the back of the range. Make sure that the neutral and the ground are not connected. If the neutral and ground are connected, the GFCI will trip.
The neutral and ground are not supposed to be connected on the connection block with a four wire cord and ground going all the way to the panel. Some installers aren't properly trained and will leave the neutral and ground connected on the back of the range.