Hi Trevor,
You might know this already, but RingElement // Matrix calls Matrix // Matrix which in turn calls Matrix // GroebnerBasis, meaning that you've probably found a gb computation that takes half a terabyte of memory!
There are other strategies for computing Grobner bases and then calling x // G, but it depends on your ideal. A different monomial order could help significantly, but it really depends on your situation, especially if the ideal is combinatorially defined, but I'm not an expert.
For instance if you can prove that the ideal is already a Grobner basis in a different monomial order, then you could just define your ring in that monomial order and call x // forceGB gens I.
Hope that helps!
Mahrud