Apple doesn't offer a native way to use your Mac as a Bluetooth keyboard for your iPhone. But there are third-party apps that let your Mac act like a Bluetooth keyboard for an iPhone. You mentioned the "Thing app from LG"—that may not be the right tool, as it’s more for LG TVs. What you’re likely looking for is an app like:
These apps let your Mac masquerade as a Bluetooth keyboard and send keystrokes directly to your iPhone.
For example to use Type2Phone:
- Download & install Type2Phone from the Mac App Store or developer website.
- Turn on Bluetooth on both devices.
- Pair your iPhone with your Mac using Bluetooth.
- Open Type2Phone on your Mac, and select your iPhone from the devices list.