Programming (flashing) MachXO2 chips through JTAG
Removing R5 to R8 and connecting an external FT2232H should actually work. Can you check the connections once again? Chances are that you are right and that the TinyFPGA programmer does not recognize the FPGA model. 

I dont know how the MACHXO2 differs from the MACHXO3 in detail besides the generally lower density but the MACHXO3 is programable via JTAG, SPI and even I2C. The latter is slow but requires only 2 wires + VDD and GND which is nice if the number of pins is limited (but one has to make sure that the dual use pins are not driven by accident as this can hinder the programming via I2C).

Generally speaking the breakout boards typically feature a FTDI chip so one can simply use any cable/board which has a FTDI on it. E.g. I successfully used the ft2232h mini module to program my MACHXO3s and ICE40s. It is recognized as valid programmer by lattice diamond as well.