Thank you for using Apple Support Communities. It's my understanding that when trying to record the screen on your iPhone, you're not getting the microphone option when firmly pressing the recording button. I'm happy to help.
I recommend reinstalling iOS with iTunes. This step can resolve possible software issues. When reinstalling iOS, iTunes will completely remove your current software and install a fresh version. While iTunes will try to do this without removing data, it's a good idea to make sure you have a recent backup before trying this step.
How to back up your iPhone, iPad, and iPod touch
Use the steps in the link below to reinstall iOS with iTunes:
If you can't update or restore your iPhone, iPad, or iPod touch
Please let us know if the issue persists.
Have a great day!