Just thought I'd do a follow-up on this.
I ended up hiring a locksmith to get my lock modified so it could have two keys in at the same time. The process was expensive but I was finally able to install and start using my sesame lock.
Sesame works fine with a lock like this. It turns my key to lock and unlock the door without issues. There is, however, one problem with this system. If the door is locked with the key on the outside, the handle positioning for Sesame gets messed up. Sesame displays the door as unlocked (even though it's locked) and can no longer unlock the door. Only the key can. An easy fix for this is to manually turn the Sesame handles until the door actually locks/unlocks and this rests the handle position on the app.
This is obviously a hardware limitation with such a use. When you lock the door with a key, the inner key does not move and confuses Sesame. More key turns are required from the inside to unlock the door which are not according to the handle positions that were memorised on the app. A simple fix to this would be to allow the app to manually turn the handles as much as one wants (for example, by turning the handle with your finger on the app in the direction you want.)
For now, I simply try to avoid locking the door with my key.