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Ahhh. You're not trying to turn Bluetooth on and off. You're trying to use Bluetooth to control a device. That makes sense.
One option would be to use an ESP32, which has Bluetooth built-in and can be programmed using the Arduino IDE. I don't recall the name of the app right now, but there is an Android app – system really – to do just this.
Solid state relays usually have a range of voltages they will operate from EG somewhere in the specs it might say 2 - 5V - see if you can find that for your relay.
If it is limited to 1.5V, you just need a series resistor to drop the excess voltage from the 2.5V.
But, to make it delay the 1 second, you will need to add extra circuitry between the Bluetooth module and the relay - that could be as simple as a resistor/capacitor/transistor combination if the 1 second timing is not super critical.
At least some HC06 modules have a "state" pin, that indicates when they are connected. This state pin can be used to activate a relay module like this one.
One question is the "state of the state" pin when connected (sorry, couldn't resist). It may be either high or low when connected. Some relay modules can be configured either way (so the relay can be off when disconnected), but that can be sorted out by using the relay's NO (normally open) or NC (normally closed) contacts.
I guess I should let you guys know that you are talking to an electronics idiot. I'm trying to keep this thing simple so I can build them at my kitchen table. If I can activate a solid state relay with the status light I think that may be the easiest route to start with.
If the state pin alternates between on/off when unpaired and sits at a steady voltage when paired, then a voltage comparator circuit will be needed to provide an output in the paired condition.
To electronics people, a comparator circuit is nothing special, however, it is going to mean a few more components and the ability to wire it up correctly for you, plus the need to fine tune it (one thing you will find in electronics - theory and actual operating hardware can be two different things).
But we need to first find out if how the status pin operates can be changed on the bluetooth module to see if the above is actually needed.
This would require (hopefully simple) programming skills.
With a bit of luck, somebody who has used these modules might drop by this thread to let you know if it is possible.