Using a transistor as a switch between RX and TX
This would work well, MAX4544. Have the default position be COM to NC with a pulldown on IN, when the Arduino sketch wakes up it can drive IN to High to connect COM to NO and enable the BT connection.

https://datasheets.maximintegrated.com/en/ds/MAX4541-MAX4544.pdf

Or one of this if you want to switch both in the same package

https://datasheets.maximintegrated.com/en/ds/MAX4528.pdf