Perhaps I don't understand the problem, but the SLAU319 pdf says in Section 1.2 that the F437 can be programmed via BSL with the BSLDEMO software. That's a Windows command line utility. All you would need is a USB-to-Serial converter module and its driver, something like the CP2102 or one of the FTDI parts, and a way to connect to the right pins.
I believe the F437 has dedicated JTAG pins, which means you would need the signaling pattern described in Section 1.3.2. The latest version of BSLDEMO from TI has an option to invert the DTR and RTS lines coming out of the adapter to get the polarity right, but if that doesn't work, I have a version that lets you invert the lines individually. You would need to run the program without connecting to the F437, and see on your scope whether the pattern on DTR and RTS is correct. Just make sure the adapter you get brings out those two lines.
Then Table 19 gives you the Tx and Rx pins used by BSL.
The big problem with using BSL is if you don't have the right password, it will erase the entire chip, including calibration data, and effectively destroy it. So make sure you know what the vector table contains, and have that in a TI-TXT password file to include in the command line. Also, if you're flashing new firmware, that has to be in TI-TXT format too.
I'd suggest you take a very good look at SLAU319, and maybe practice with a cheap DIP part like the MSP430G2553 (after using JTAG SBW to save a copy of its INFOA in case you erase it).
I realize that BSLDEMO is old, and officially deprecated by TI, but they continue to update it, and you don't have to buy expensive gear to make it work.
Oh, and I also have a VBScript that converts Intel-hex to TI-TXT. All of my stuff is available on Github. It was written with the G2553 in mind, but it should be the same except for the initial signaling pattern.
https://github.com/gbhug5a/MSP430-BSL