PIC programmer problems reading and writing program configuration memory
So I managed to read and write the configuration word by sending an unprogrammed data word (i.e. writing a word of all ones or 0x3FFF as a word is 14-bit) just after the load configuration command and before incrementing seven times.

It turns out that this is the typical way of writing the configuration word and yet the programming specification didn't bother mentioning it and I actually read the programming specification of a handful of different models until I found this information.

So, my advice is if you have a similar problem and can't find the solution in your data sheet or so, go check the similar models' documents and maybe you can find a hint or two to solve your problem.