It's a logical contradiction in a simple way:

If you assert that 'moral relativism is true' that itself would be a relative claim which would undermine the assertion as a universal position.

Further, you end up in a position of equivalence of all claims and generate major contradictions.

I will go out on a limb on the universal moral values claim and I think again it's pretty simple:

The existence of even the most basic objective truth, such as 1+1=2 would imply by its existence a frame of reference by which all claims can be evaluated against. This constrains the scope of relative claims, as you pointed out, but also implies an actual truth or set of standards with which you can evaluate anything.

It's not clear what universal moral values looks like, like Epicurean pleasure/pain or maybe it stems from thermodynamics, or even just mathematics itself, but it is possible that it could be discovered.