Welcome to the Community!
Did you check with Telstra if the modem that you received from them will work with another provider?
If yes, then you need to check on what configuration are you going to use with your third party device.
Please contact iiNet on their support page, https://www.iinet.net.au/contact for further assistance.
Though, TPG and iiNet are under one umbrella, we still operate independently of each other.