It may be interesting to consider further constraints or specs for any solution(s), but I think we should still retain the originals (as best we can understand them).  One of these is that the "total" (which I understand to be the SUM, not just the SET) is the "lowest valued base 10 square number available."  this cannot be less than 65 (sum of 11..15, the lowest five distinct bases), so the next lowest square is nine squared = 81.  As output listings have shown, there are many solutions with total = 81.  I merely changed the highest (16) of my original list (i.e. after Charlie's clue) to find one (11-12-13-14-31) which came to 81.  Dej's extension is interesting, but requires ignoring one of the criteria that WERE given (lowest square for sum of bases).  I did consider requiring 10 unique digits in the re-based values, but thought that would have been clearly stated (e.g. as in alphamerics) if it were to be required.

The graphics were attractive, but the first stage of a solution was so tedious I almost ignored the whole: creating the second grid, to find which sums of orthogonals of the 36 cells were squares (and then double and triple checking those).  Since this transformation itself played no part in the rest of the puzzle, we could simply have been given the set of squares (16-36-49-64-100) and posed the actual task.

I thank brianjn, the proposer, for his comments on the evolution of this perplexing task.