Assuming this is the matrix you emailed into us, the determinant is not just "very small", it is essentially zero. 1.05e-101 is not "very small". it is zero.
Yes other packages may invert the matrix, but the result they give is a matrix full of extremely large numbers. This matrix is unworkable, and will disrupt any further analysis you do using it. The fact that it is only "part of some transient calculation" is irrelevant - were you to use the result from Excel or Stata all of the calculations based upon the inverse would be meaningless.
I think you're approaching this problem in the wrong way. Rather than trying to get EViews to invert a singular matrix for you, you might be better off working out what is making it singular in the first place (which is essentially an econometric/economic problem, not a software problem).