Yes, the C Wire needs to be connected when the thermostat is paired for it to be repeater mode. One possible way is to do a factory reset on the device. This should make the device unavailable to SmartThings and also exclude it from the hub.
To be safe, disconnect or power off your hub BEFORE doing a factory reset because some device send an event to the hub indicating a factory reset and the hub automatically then removes the device from it’s list of devices, powering it off or disconnecting will ensure it doesnt’ get that event.
Once it’s reset, power on your hub, goto your thermostat device page, click on the settings gear icon, scroll down and click on “Replace” device. It will search for the existing device and find it doesn’t communicate and will give you an option to pair a replacement. Now ensure your C-wire is connected and start the pairing process.
This should allow you to re pair the same thermostat without excluding it. Again, I’ve tried this once and it worked here in the lab but I didn’t have any attached apps to it, it was just a PoC so you may want to back up your apps/pistons to be safe before trying it.