Or is it possible to overcome the fault, since the code isn't in the monolithic part of the kernel?
It is possible to be overcome from here:
You should also give some thought to where you do your module experimentation, development, and testing. We have done our best to make our example modules safe and correct, but the possibility of bugs is always present. Faults in kernel code can bring about the demise of a user process or, occasionally, the entire system. They do not normally create more serious problems, such as disk corruption. Nonetheless, it is advisable to do your kernel experimentation on a system that does not contain data that you cannot afford to lose, and that does not perform essential services. Kernel hackers typically keep a “sacrificial” system around for the purpose of testing new code.
A fault in a kernel module may bring down a process if the module is running in process context, (e.g: due to a syscall), or it may bring down the system, if the module is running in interrupt context. Faults while registering/unregistering the module may hang the insmod/modprobe process, and of course you won't be able to unload the module. Infinite loops may leave a core/CPU running at full throttle, and unavailable to other processes.