My wife had a brainstorm and we tried cleaning the tub with a dilute bleach solution, letting it soak overnight, and it worked very well. The tub still needs a cleaning, but the bleach loosened the worst of the dirt.
I'm viewing this is a pre-step to a proper scrubbing, but one that will make the actual cleaning easier.
If we were to do this regularly, say every few weeks, could this be bad for the pipes?