The best option so far, and one that works really well, is to use QuickTime. Until now I used to believe that QuickTime would not allow you to record both the sides of the conversation, but it is possible. I've pasted the link to this video below which should explain things in detail, but I'll sum it down for you:
- Firstly, you'll need a software capable of recording computer playback audio (I use Loopback).
- Open Spotlight Search (using CMD + Space) and type in "Audio MIDI Setup". Open the app.
- On the bottom-left side of the menu, click on the '+' icon and select "Create Aggregate Device". It basically allows you to use multiple input/output devices at the same time.
- After that, just to the right of it, it'll show you a list of your input/output devices. Under the heading "Use", check the box for the ones you'd like to use during the screen recording on QuickTime (For the purpose of FaceTime, it should basically be whichever microphone you're using and Loopback. Make sure you select Loopback.) Don't alter any other settings.
- Normally when you create an aggregate device, the Mac automatically names it "Aggregate Device" but you can call it whatever you need to.
- Close the Audio MIDI Setup window.
- Give the Mac a few seconds.
- Now, go to QuickTime and under the "File" heading, select "New Screen Recording".
- Next to the Red Recording dot, click on the downward facing arrow, and under "Microphone", select "Aggregate Device" (or whatever you named your configuration).
- That's it. When you record and playback the screen recording, you should have the screen recording, the playback from the laptop as well as the input from your side.
I would suggest testing it before the call. Also, the reason I suggested to wait for a few seconds after creating the new configuration is because I tried it myself and the new Aggregate Device does not instantly show up.
To make sure that the Aggregate Device was created, you can even go to the sound settings under System Preferences, go under the "Input" heading, and check if your Aggregate Device configuration shows up there.
Here's the link to the video that helped me:
https://www.youtube.com/watch?v=Q2d4JEwMCCQ
Let me know if you face any issues.