I use a small fountain pump from Lowe's $12.00. I circulate the water from the bucket through the nipple run and back to the bucket. Solved all my freezing problems. The sketch does not show the return it was added after.
What kind of water cups did you get that hold an entire cup of water? Can you post a pic or give us a link to see them. All the cups I am familiar with and have experience using also work with a small valve but they are activated when the birds drink and push on the valve with their beak. A tiny amount of water is released each time they take a drink and a little is left in the cup for the next bird to see.
I think you would be much better off using the horizontal nipples like Rich sells and building your system with a return line so you could recirculate the water. If you were to put a small stock tank heater in your reservoir and circulate the water you would never have to worry about your system freezing and with the slightly warm water flowing behind the nipples they would never freeze either.
The thought of using a 250 watt heat lamp all winter to keep a cup of water thawed scares the FIRE out of me and then you have to consider the electricity used would be way expensive. You would save enough on your first bill to spend $21 for 10 nipples and then the next month you would save enough you could buy the pump and heater and never have to worry about your system freezing again.
With an auto fill system and circulation pump you need very little water in reserve. That would allow you to use a small heater like a birdbath heater or fish tank heater. These would seem to be cost effective not only to buy but to run. Seems that a setup like that would stand up to very low temperatures. Here in Live Oak, FL we only get about a month where we have to worry about pipes freezing. After replacing broken pipes three times the pump solved my problem. I have not tried either one of these heaters but would love to hear from those that have so I could pass it on.