We used to do something similar to you when I was in RCI at Tooting, except that we performed tube IAT strictly at 37oC, with warmed NISS for the washing process. As far as this is concerned, I could never see any difference between having the red cells initially suspended in NISS or LISS, except, of course, that the incubation time was shorter with LISS (although we used to bring the reactants to 37oC before the initial mixing). Again, like you, this was mainly for pregnant women.
After many experiments trying to get the CAT reaction chamber strictly up to 37oC BEFORE the addition of the reactants, and failing miserably, we abandoned the idea, as we could never quite get to 37oC. Unfortunately, "cold reacting" antibodies can (and do) sensitise their cognate antigens extremely quickly, but are much slower to dissociate, so we were never quite sure whether the anti-M was reacting genuinely at 37oC or not.
I think you may have made a typo. There are no enzymes involved, because of course, the M antigen is papain sensitive, by the columns have a slightly low pH, and this, together with centrifugation and the difficulty in keeping the reaction chambers at strictly 37oC all mitigate towards the possibility of a false positive at 37oC.
I hope this bit of rambling answers your question sufficiently, but, if not, do not hesitate to get back to me (or anyone else, come to that!).