Hi there & welcome to the boards!!!
Looking at your schedule, I would say that the naps (or should I say timing of) is whats causing the night wakings.
At 5 months, she should be able to stay awake for 2-2.25hrs so you want to aim for something like this:
6:30 - wake & feed
8:30 - morning nap (1.5-2hrs)
10:30 - wake & feed
12.30ish - afternoon nap (1.5-2hrs)
2:30 - wake & milk
4:30-5pm - catnap (45mins)
6.30 - milk
7pm - bedtime
11pm - DF
Possibly another feed in the early hours of the morning (4 or 5am)
The night wakings I believe are because her afternoon nap is too short, so the catnap is in the late afternoon rather than early evening & she is then awake for 3.5hrs before bedtime. Thats the av A times for an 8mo so I would think she is extremely tired by bedtime.
How does she fall asleep for naps? Are you able to stay at home for the afternoon nap to work on it & try to extend it beyond 40mins?
The schedule is the ideal for a 5mo but naps don't have to be as long as 2hrs. But it would be more ideal for the afternoon nap to be longer so if you wanted to, you could mess around with the nap length & maybe wake her from the morning nap 1.5hrs later so that she is more tired come afternoon & might sleep longer as well. The catnap is a freebie so can be taken in the stroller/buggy/sling/car. That was the 1 nap I didn't really care much about (in regards to being in a cot) so if I had to run errands, I would wait for the evening.
See how you go with that & let me know if you have any q's
Layla