I would like to see a number > 3.5% (since 1990) http://www.neighborhoodscout.com/il/chicago/rates/
If below 3.5% I would probably pick the $650k option if I could ensure that I had the correct school district, bedroom and bathroom for au pair, let kids share? and lot size to expand from your "Alternative to buy a smaller place in the city with no au pair option. This would cost $650k to $750k."
I completely understand paying for big mortgages, small commutes and paying an extra $100k+ for the right school district. However, if it is borrowed money, I would tend to not try to get everything at once... just focus on the things that can't be easily deferred (school district) but edge into the ideal and then allocate bonus etc to extensions or improvements.
Sounds like you have no issues affording the $850k if that is what you want... but I would get more value from the other $200k from e.g. another rental property that I would expect to appreciate at > 3.5%, cost me zero cashflow and 100% ownership in 15 years. However, I also enjoy my 15 year old car - so I think your question is more about how you find value