Interesting idea, but the TEA algorithm appears to need four, "^" operations per pass through the loop, so anything from 32 to 128 depending on the complexity chosen.
That appears totally impractical on an 8 bit CPU?
Event the SHA256 is going to take a very long time (in CPU cycles) on an 8 bit device.
A much simpler access control system exists for a low power device that has a direct internet connection - port knocking.
It is extremely simple, very secure and the device is invisible to probing.
I'd say that any important device incapable of full HTTPS security should not be connected to the internet anyway, it should be on a separate, isolated or firewalled connection to a gateway system of some sort that has full protection / encryption.
Personally, I think the whole idea of using outside resources for devices with inputs and outputs within the same home or premises is fundamentally wrong - especially when it comes to power control, HVAC or security!
External access is a useful feature, but no part of the system should fail if outside connectivity is lost.