Since the chip is socketed, the easiest way to read it would be to pull the chip and insert it into a programmer. (I assume you have one, and intend to use it to program the boot loader as well?)
This would still fail if the chip is read protected. The fact that the board is cheaply produced does not mean that the manufacturer did not care about their IP. Maybe even more so, if this device was made in large quantities, and it is easy to copy the physical circuit board.
But I don't think that you need the original ROM contents in order to re-use the board for something else. You should follow the PCB traces to see which processor pins are connected to which connector pins and other hardware on the board. That's much easier than to analyze the original program.