Shutting down the APU straight away should not affect anything unless you have other faults present (Eng failure not included). There are a quite a few faults listed in the DDG under Det FIRE APU. For your scenario shutting it down when the checklist asks you to is ok. At that point in the Checklist I would probably refer to the DDG and have a discussion ablut further affects or ground contingencies.. There is a note that you can still use the APU once parked at the gate/stand with a fire guard near the tail of the aeroplane to monitor the APU.
Lots of good advice from other posters especially about not blindly following a checklist.