I think you may be engaging her too much with the picking up, hugging, singing, kissing, etc. You've created a new ritual that is comforting in its regularity and predictability. So the middle-of-the-night ritual is just as involved as the pre-bedtime ritual, only quicker. She's not self-soothing at all because she's counting on all these things from you. There's too much interaction, too much touching, even though you are not cuddling for any length of time. So it's too little interaction to comfort her and soothe her (so she wants more holding and singing).
Try not picking her up and not talking to her. Just a back pat, "it's night night" or "it's sleep time" (whatever your code words are going to be) and out the door. If you really really feel you need to check the diaper, just reach in to the diaper area and check that, rather than hugging to get close enough for a sniff!
This extensive interaction is also engaging your brain too much, which is one reason you are so awake for so long. You're wondering how long she's going to fuss, you're thinking about how tired you are being pregnant and a working mom of 2, with no husband at home because he's working that night shift.
So, night 1, let her fuss for 10 minutes, then go in and say your code words in as soothing but firm a voice as possible., pat her back but do not pick up or hug and do not do an extensive massage, and leave. Then wait at least 20 minutes before going in again. No pat, just code words. Night 2, wait 20 minutes the first time, 1 pat, then out, then 30 minutes before going in again. Night 3, wait 30 minutes.
I know you don't want her in a wet or poopy diaper all night, but if she were to sleep through the night, would you still be going in to check her diaper? Or would you just deal with it in the morning? If the latter, then maybe you shouldn't even be checking her diaper unless she is complaining or clearly soaked through.
You really want this totally taken care of by mid-April so you can get several months of sleep before the last trimester when you a) need rest and b) can't get comfortable!