Your SIM card just grants you access to any particular network, your phone can still be tracked off of its serial number or IMEI as other towers in the area, regardless of whether or not they are your carrier, will log the devices that pinged the tower.
This MIGHT be ok with a burner phone if you paid with cash, but they could still pull data from in store security systems of the purchase by tracking the phone to the point of sale with its IME, if they were sufficiently motivated.
It might work, But it’s a gamble I’m not willing to take.