The problem is not with your router. I was experiencing the same symptoms with my phone and called TM Support -- they indicated that this is a known issue with the latest firmware update.
When the phone is running on battery and goes into idle mode, wi-fi calling disconnects. The temporary workarounds are to either 1) disable the screen timeout (which will obviously eat up the battery) or 2) keep the phone connected to a charger.
The rep indicated they are working on a fix -- no ETA for that, however.
Since the TM signal is weak at my home, and wi-fi was about the only way I could use my phone at that location, TM sent out a free Cel-Fi unit, which has helped dramatically. Cel-Fi does use cell minutes, however, unlike wi-fi. Accepting the Cel-Fi unit also puts you on a new 2-year agreement, but since I was only a few months into my current one, this was not a big deal for me.
Thanks for info. It is good to hear it is a known issue, although I'm not too happy there is no known solution.
There does seem to be a router component to this problem, though, since I did not experience this problem with the latest firmware while connecting to the WRT54.
I'm going to hook up the WRT54 as a wireless access point running alongside the WNDR3800 and make a "WIFI calling" SSID. If that doesn't work, I guess we'll have to wait for a firmware fix. Luckily my cell reception at home is not so poor that I can't live w/o WIFI calling. But clearly this is not the case for everyone.
A brief update:
I've confirmed that there is indeed a router component to this problem. I tried out an ASUS RT-N56U (a simultaneous dual band wireless N router like the Netgear WNDR3800) and have had no problems with the wifi calling droppingwhile on battery power.
There are known "wifi disconnect" issues with the WNDR3800, although I was not experiencing those on the latest FW. I was only experiencing "wifi calling disconnects" on this router.
Anyway, I'm returning the 3800 and sticking with the ASUS. It seems in general the ASUS is less buggy.