What you need to have is a Bluetooth adapter with MAP (Message Access Profile) support. Your iPhone will treat all messages from the MAP protocol as if they are from the Messages app. This means it will automatically route your SMS as iMessage if possible (you have no say on what the iPhone decides to do, however). As a bonus, you can also use email addresses as recipients with MAP.
A good place to start probing is the WT32 or WT41u module from Silicon Laboratories. It supports MAP, although it looks like the module supports receive-only [1]. I do not know whether you can hack blueZ to support MAP. I've tried to look at it and I don't think the MAP support for blueZ is complete but I could as well be very wrong. A Raspberry Pi 0 as a bluetooth middleman is very sweet, regardless.
Once that Bluetooth middleman is set up, you can use a public server to relay your messages. The scheme will look something like this:
iPhone <--bluetooth--> (WT41+ESP32)|(Pi0+BlueZ) <--wifi--> MQTT broker <--wifi--> your device of choice.
I am relatively confident that this scheme will work. I just don't have time in my hands to do it. So I figured I could share here. Hopefully, some good hacker will do it and publish it. Happy hacking!
1: https://www.silabs.com/documents/public/application-notes/AN...