I'm trying to set up a hobby project and I'm wondering - if my device uses Bluetooth Low Energy module, will this device show under the list of Bluetooth Devices under Settings for iOS? Or will my app have to search for this device specifically before connecting? If it doesn't show in the Settings, what type of Bluetooth module would I need for that?
No, you will not see your LE devices under Settings for iOS. See LE best practices - your app should search for the LE device.
You only see the BT classic devices in that settings. However, only few protocols could be used without a required MFi chip, which is out of your scope for your Hobby project.
Use The CoreBluetooth on the iOS side for your app.
UPDATE per @Brejuro request in the comment below:
MFi coprocessor is required to be in your custom HW, which wants to communicate using the majority of profiles/protocols defined for Bluetooth Classic (not LE), e.g the pure RFCOMM. Coprocessor is used for the initial authentication - then, iOS knows that your custom HW is certified and allows to use it.
However, Apple will ask you for big money and your device needs to pass through Apple certification - that's why I told that this is out of the scope for any hobby project. You will also need to sign very restrictive NDA. However, you will get very very good documentation describing many details.