Thanks to some support on the Discord server, I managed to pinpoint the issue. It's tied to modern CPUs and multithreading. You can fix it by either a) launching the game with the launch parameter +set sys_smp_allowed 0 or by b) setting the CPU affinity for the Plutonium launcher's .exe to only CPU 0.
a) works fine on vanilla WaW, but only works on Plutonium if you're using an offline mode bat file. b) works on Plutonium perfectly fine, but you have to either manually set the affinity through the task manager every time, or you have to create a shortcut to automate it.