your stm32l433rct has a different amount of flash and ram, so I would not expect the l473 binary to run out of the box. I already wrote ports of Mecrisp-Stellaris for many STM32 chips, they are quite similar, changing memory constants and perhaps touching UART init should be enough to get it up and running. Are you fine with ARM assembly, would you like some pointers on how to do, or would you rather wish that I take care of that, changing the constants according to datasheet and giving you hopefully correct binaries for testing ?
Actually, there are two sets of TX/RX lines, one pair for the terminal into Mecrisp-Stellaris, which includes bitstream and loads the FPGA, and another pair between ARM and FPGA, which is bridged in software into the main terminal. Do you wish to use both Mecrisp-Stellaris and Mecrisp-Ice, or are you interested in Mecrisp-Ice only ? For testing, you may use the bitstream standalone with the official Mystorm loader and directly connect to the FPGA terminal wires.
PS: Yes, Mecrisp-Stellaris will greet you and reply with OKs (or error messages) when up and running.