That "sine" looks like one or more of the bits, including the most significant one, are faulty (ie held constant). The question is if that fault is in the DAC itself or in the FPGA (or otherwise outside the DAC chip.) There's a DAC near the voltage reference. Is that DAC setting the voltage reference, or is it using it? If the latter, is there a second voltage reference near the output DAC you can try doing something about?
The reversed polarity of the reference TP's are obviously suspicious. Maybe try tracing that back through the components and see if gives you something useful. My theory is that they're both generated from the same reference somewhere, and one is inverted through the opamp. But the actual reference is blown, such that it output a voltage of the opposite polarity compared to the expected value. And then that creates the faulty test point values.
But yeah, this is starting to look like beyond economical repair. Maybe one of the in/outputs of the unit received a large negative voltage which leaked into the negative rail through protection diodes and destroyed everything in its way on the negative side.