When changing a battery after the initial settings have been set and the first flight is completed, will the initial settings have to be repeated with the second battery or can I just take off knowing the settings haven't reverted to some default?

The settings will remain. Only after firmware updates you should check everything again because some might reset to default in the process.

I always check the RTH setting before each flight just to make sure, best to be safe then sorry

Wachtberger is right, theoretically, changing the battery will not set the settings back to default.