There is no need for a second motor. Any control scheme can be connected to a single motor - all you need is an external relay, or relays, to provide the changeover contacts.
If you use two timers, they could both run off the same battery. This could be the same battery that powers the motor.
There's not enough detail on that timer to tell whether it's suitable or not, but I don't think a single one will do it - it's designed to time "feed times" with durations up to 30 seconds and I think it only has one output.
I've been thinking about this project and I realised that we have to consider what will happen if the door gets stuck for some reason. With the arrangement I drew up, if the limit switch isn't reached, the motor will just keep running. This could damage the motor or the mechanical parts of the system, and it also wastes power, which is important for a battery application.
I think it would be better to control the motor from one output that activates in the morning, and a second output that activates in the evening, but the outputs only activate for, say, one minute. The alternative would be some extra timing devices to make sure the motor doesn't run for too long, but the first idea is simpler.
I haven't found a timer that has two separate outputs that can be configured to pulse one output on for one minute in the morning, and pulse the other on for one minute in the evening, but you could use two of these generic timers:
http://www.ebay.com/itm/LCD-Digital...-Time-Relay-DC-12V-16A-USA-Ship-/321127095779
I'm really hesitant to recommend anything that's cheap and Chinese and sold on eBay, but it looks like it would do the job. You would use two of these, powered from the same 12V DC battery. One would be configured to turn ON for one minute in the morning, and the other would turn ON for one minute in the evening.
Those timers have just a single normally open output contact, and you would need to drive a relay from each of them to provide the extra switching needed to reverse the motor. I assume you will want these relays to be panel mounted, with screw terminals for connections. Here are links to a relay and a panel-mount socket on Digikey, so you know generally what you're looking for. The relay and the socket are NOT COMPATIBLE with each other; I can't find a suitable relay with a compatible socket on Digikey! These are just to give you an idea of what you want. You will need two of each.
Relay:
http://www.digikey.com/product-detail/en/XT424012/PB1194-ND/1841562 (around USD 6 each)
Socket (for DIFFERENT type of relay):
http://www.digikey.com/product-detail/en/27E487/PB642-ND/678275 (around USD 7 each).
You should be able to get suitable relays and sockets from a local electrical supply company. The relays need to have a 12V DC coil and DPDT contacts. The contacts must be rated for 12V DC or more, and several amps - I suggest at least 5 amps.
These relays can be controlled from any timer or timers that have a relay contact output with two pins or more. I suggest you look for a good quality timer that can turn ON for one minute at a programmed time of day, and get two of them, or ideally, a single timer with two outputs, that can turn each one ON for one minute at different times of day.
If you can't find anything, you could buy two of those units on eBay that I linked to. I don't recommend eBay products generally because they're often poor quality. Buyer beware and "your mileage may vary".
Actually, two of the Wildgame Innovations TH-DT timers you linked to would probably be suitable, as long as the door doesn't take longer than 30 seconds to open or close. They are about three times the price of the eBay units though. Your call.
One last thing I forgot before. The motor should have a diode connected across it, to suppress the voltage surge that it will generate when it's turned off. Without the diode, this voltage surge can cause arcing in the contacts of the relay or the limit switch, which can damage it over time. I will include a diode when I draw up a new diagram, once you've decided on what timer(s) to use.
