Well, I've just discovered that alongside "programmer.exe", there is also "programmer_cli.exe", which lets you do everything via command line, so the clicky-button problem is solved.
But now I'm thinking, whether I really need to be swapping out the Gowin JTAG adapter and the J-Link JTAG adapter - I don't leave them both physically connected to the FPGA JTAG port at the same time out of concerns for damaging something, but if all the JTAG lines are open drain then that shouldn't be a concern, right?