Just a thought - your need is to transfer data at 5MB/s (or a tad more) from some FPGA to a PC.
How many IOs would you actually have available? QSPI would require a strict minimum of 5 signals, preferrably 6 with a /CS signal, and this, only for one-way communication (if that's what you are after). For two-way comm, you'll need 4 more lines - you could also change direction dynamically, but QSPI per se doesn't support that, that would be a bit cumbersome. So for 2-way, that would be up to 10 lines.
With just a couple more, you could use some FTDI chip like the FT232H instead of having to use an extra MCU, write software for it, etc. The 245 asynchronous mode would get you over 5MB/s with no issue, and that comes with certified drivers. I know it's popular to bash FTDI, but it would just get you there in no time.
If you still don't have enough IOs foir the 245 mode, you could use the FT1248 mode (if IIRC) of the FT232H which can be used as some sort of SPI with a bus width of 1,2,4, or 8. In this mode, you can still get transfer rates up to over 20MB/s.
And an FT232H is much cheaper than one of those beefy MCUs (like the TM4C1294) you were thinking about.
Just my 2 cents.