Can a GFCI plug on the same circuit as a range cause a fault for power getting to the range plug?
You can't do that.   The 240v power to the range must go only to the range.  You can't fork off one leg of the 240 and a neutral to also serve a 120v plug.  Of any kind.  If you need a 120v plug there, you need to either run a new circuit, or add an outlet to an existing circuit.

That, however, does not explain why the GFCI is tripping.   The GFCI would not know it was applied illegally - as long as it sees 120v across its line terminals, it is happy.   The fact that it is tripping suggests to me that either it is defective or you have connected it to 240v.  Or it is working properly and you are plugging in an appliance which has a ground fault, such as a 3-light electricity tester.