Constant connection with WN3000RP?

I'm not aware of any setting on the WN3000RP.  Are you sure your iPhone stays connected to your Uverse router?  I Googled the iPhone's Wi-Fi behavior and AFAICT iPhones will put their Wi-Fi into power saving a few seconds after going to sleep (e.g. link).  I confirmed this with my own iPhone 6S connected to my R7000.  See ping output below.  The request timeouts indicate when the iPhone stopped responding shortly after going to sleep.

There is one exception to the above.  You can prevent the iPhone from turning off Wi-Fi by keeping it plugged in.  

I've read that turning off Cellular data may keep Wi-Fi enabled, but I found that this did not work. 

$ ping 192.168.1.33

PING 192.168.1.33 (192.168.1.33): 56 data bytes

64 bytes from 192.168.1.33: icmp_seq=0 ttl=64 time=40.924 ms

64 bytes from 192.168.1.33: icmp_seq=1 ttl=64 time=63.159 ms

64 bytes from 192.168.1.33: icmp_seq=2 ttl=64 time=81.448 ms

64 bytes from 192.168.1.33: icmp_seq=3 ttl=64 time=101.025 ms

64 bytes from 192.168.1.33: icmp_seq=4 ttl=64 time=19.887 ms

64 bytes from 192.168.1.33: icmp_seq=5 ttl=64 time=39.052 ms

64 bytes from 192.168.1.33: icmp_seq=6 ttl=64 time=58.223 ms

64 bytes from 192.168.1.33: icmp_seq=7 ttl=64 time=78.970 ms

64 bytes from 192.168.1.33: icmp_seq=8 ttl=64 time=97.069 ms

64 bytes from 192.168.1.33: icmp_seq=9 ttl=64 time=16.171 ms

64 bytes from 192.168.1.33: icmp_seq=10 ttl=64 time=6.200 ms

64 bytes from 192.168.1.33: icmp_seq=11 ttl=64 time=54.967 ms

64 bytes from 192.168.1.33: icmp_seq=12 ttl=64 time=78.374 ms

64 bytes from 192.168.1.33: icmp_seq=13 ttl=64 time=100.766 ms

64 bytes from 192.168.1.33: icmp_seq=14 ttl=64 time=18.493 ms

64 bytes from 192.168.1.33: icmp_seq=15 ttl=64 time=38.500 ms

Request timeout for icmp_seq 16

Request timeout for icmp_seq 17

Request timeout for icmp_seq 18