You may be going about this backwards - start with the application requirements and then pick the microcontroller based on the requirements
I am an electronic hobbyist so perhaps one of the more learned members will jump in if I get part of this wrong. There are only two UARTs on the FR5969. Pins 2.1 and 2.0 connect to the Ez-FET which is a USB to UART bridge that allows both programming of the target microcontroller (in this case the FR5969) and serial communication (in the case of Energia through a window that you open up).
If you want to use Pins 2.1 and 2.0 for something other than programming and serial communication then 1) download the firmware as usual over USB using the Ez-FET, 2) remove the USB connection, 3) remove the RXD and TXD jumpers and isolate Ez-FET from the target microcontoller, 4) connect the new device (e.g. GPS) to RXD and TXD on the target side - i.e. FR5969 side, and 5) power back up. Now the new device can use hardware UART over pins 2.1 and 2.0.
Of course you can't use serial over USB to the PC now since the jumpers were removed. If PC connection is required then you can use software UART (limited to 9600 baud) or select a microcontroller such as the TM4C123 which has plenty of UART peripherals available.
I've looked at the User's guide and although the board has 2 UART interfaces, one of them is called 'Application (or "Backchannel") UART'.
If I am unable to use both UARTs on board to communicate, how is it that the combined GPS + GSM module can interface with the micro-controller using just the 1 interface?
I am not that familiar with that device but the Energia pin diagram you link to shows the following:
Pins 3 and 4 on the left hand LaunchPad header are labelled P2.6 and P2.5 as well as RX(1) and TX(1)
The programming header is labelled RXD and TXD. Looking at the LaunchPad User's Manual these turn out to be P2.1 and 2.0.
There are two UARTS available in hardware: USCI_A0 and USCI_A1.
I haven't tried this on the FR5969 but quoting from the Energia documentation on Serial: