It is not possible to use a pre-compiled .a file with the cloud compilers.
It may be possible with the gcc-arm local compiler and Particle Workbench. However, you can't do it using the build.mk file, so you actually need to modify the Makefiles in the device-os source in order to get it to work.
Success! Thanks to the link you gave me, I was able to flash the precompiled BSEC library to my Particle Photon. I had to rearrange the library a bit so that the .a file I needed was in the root, and then just like that user described, it was just a matter of editing the two makefiles:
I wasn't actually sure what I was doing translating linux directories to Windows file structure, but I was pretty sure that what I had just told it with project_root was to look in C:\Users\USERNAME.particle\toolchains\deviceOS\2.3.1, so that's where I added the BSEC library as well, and I just renamed it BSEC to shorten it. I'm not sure which directory the compiler was really looking at, so I just made sure if I made any changes to BSEC library in one, I copied them to the other.
It compiled successfully, but threw up an error when trying to use the cloud compile button, so I just ran