So it seems clear that Porter wanted "sich" pronounced that way -- but that doesn't mean any of us have to agree it's a good choice, and for what it's worth, I don't.
The use of regional pronunciations like "sich" and "fust" in OKLAHOMA! and CAROUSEL has almost always taken me out of the performance, perhaps because the performers so often hit those pronunciations hard in order to make the point, rather than making them sound natural and organic. Also, in the case of "sich," maybe this is a regionalism that was once very more common than it is today?
In any case, I stand by my personal opinion that if a singer makes a point of singing ".....who is rich but sich a bore" rather than "....who is rich but such a bore," it probably serves to mildly confuse and distract the bulk of the audience while pleasing only those who know that's the pronunciation Porter wanted :-)