Flip need not be handled on TX/Rx if you are designing a device with Type-C plug. The host end will implement flip if it has Type-C. You can route just one lane of SS from the connector to the FX3 PHY and leave the other lane on the connector open (NC).
TX_P and TX_N will not get flipped when using single lane and SS PHY is designed to internally handle differential pair flips.
If you are designing for receptacle, please use a Superspeed Mux. Shorting TX1 pair with TX2 pair, or RX1 pair with RX2 pair is not recommended at SS bitrates.
For more details refer to the KBA pointed out by Muthu.