What are a few good ways to connect multiple I2C busses?
In an ideal world you would have your other sensor on a separate I2C bus. It can be done with software, but chances are you will have to re-write the library that supports it to use bit-banged I2C.
Yes, you could use an I2C MUX to share the single hardware I2C channel, however that will both be awkward and the results will be sub-optimal anyway.
Most I2C MUX chips, including the TCA9543A, are controlled themselves by I2C, so you can't really use them in your environment, since you really need to switch it to isolate your main bus from the MCU before you switch it from slave to master, which you can't do until after you have switched it from slave to master. If you switch to master before isolating the bus you risk a collision with the main master on the bus and everything going horribly wrong.
These MUX chips are really designed for allowing one master to talk to multiple slaves that share the same addresses.
And on top of that when you've isolated the bus the main master will no longer be able to communicate with that slave board, which will either cause the whole bus to lock up or other strange and unwanted things to happen.
So have another go at using software I2C to communicate with your sensor. Most I2C communication to sensors is pretty straightforward and follows the same rough format, so it really shouldn't be too hard to implement.
Either that or consider replacing the ATMega328P with a more capable microcontroller that has multiple I2C busses.