U-Boot would be an example of a fully open source "bare-metal" program running on the MT7688.
Keep in mind though that you probably won't get the wifi radio working, without inordinate difficulty.
Also, if you do any more than learn about the chip by reading the U-Boot sources, you should consider if that means your work will be derivative enough to need to be under GPL; however, such should not be incompatible with an academic research project.
But take a step back, there are plenty of MIPS processors of various capability out there, many potentially a lot easier to get your code into and with more public documentation. If you need a system with a lot of DDR memory, then an MT7688 might be an economical choice - but even there, you probably want one without a shield can, modified to socket the SPI flash or at least let you get a programmer on all the pins while the processor is held in reset.
And you probably want JTAG working well. I was able to use it crudely to rescue some systems earlier this year, but there were incompletely understood parts that had an imperfect success rate and were "try until it works" - for kernel development, you'll want something where you can really reliably debug.
Given you don't have an application goal, the difference of a few dollars in module cost isn't what matters - pick instead something with the best documentation and the best understood debug interface.
That won't be the MT7688.
Realistically, your most efficient path will probably be to do all your theoretical work on a simulated machine, where you are completely free to stop it, examine anything, snapshot or restore a core/state file, etc. The only thing you really need hardware for is developing the low level drivers for hardware.