The thing about dosing at 3 AM is that they start needing it at 3 AM and if you give it to them at 8 PM or even 11 PM it may have already worn off by 3 AM.  Also, growth happens a few hours into their sleep so if you dose at 11 PM you’ll interfere with growth.  Why don’t you try 3 AM for a few nights.  Maybe she’s more likely to be fussy at 11 than at 3.  Maybe she’ll be so fast asleep that she won’t totally wake up which is what I think happens with my son.