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Current status as of 7/17/17 - after a few more emails and a phone call, I am sending my phone in for repair. I'll update this once I get it back in a few weeks or so.
Quick synopsis: after turning on data saver, MMS can no longer be sent or received if phone is connected to WiFi. Turning off WiFi and being connected to cellular data allows MMS to work. Turning off data saver once it is turned on does not reverse the issue. SMS works the whole time.
More details: So this all started after I turned on data saver a few weeks ago but I wasn't quick to say that was the cause of it. After turning on data saver, I was unable to send or receive MMS messages while connected to WiFi or when phone was on standby while driving between WiFi points, so if I went to work where I automatically connect to WiFi and didn't turn on my phone screen, I wouldn't receive any MMS messages. Even turning off data saver did not allow me to send MMS while connected to WiFi.
After factory resetting the phone and before installing anything except for preinstalled app updates (also disabled AVG and Facebook), I tried sending MMS while connected to WiFi and it worked fine. Right after making sure my email received my MMS, I then turned on data saver. Previous issue then started. Turning off data saver still did not allow MMS to go through again. Redownloading MMS/carrier settings did nothing either. I also tried allowing unrestricted access to Google Play Services, phone, email, Gmail, messaging, and SIM toolkit (did not see any other potentially relevant services) but that did nothing as well. The only way I am able to make MMS work again is by factory resetting the phone again and not turning on data saver.
Somewhat related: using the toggle button when the top of the screen is pulled down to turn off data saver does not seem to actually turn it off. When I go into settings after turning off the toggle button, settings still says data saver is turned on.
Phone details:
Xperia XZ model F8331
Carrier: T-Mobile
Country: United States
Running stock firmware with Android 7.0 build 39.2.A.0.442 (latest update according to Xperia Companion and software update)
Phone has never been rooted/no custom firmware has ever been installed or attempted to be installed on it.
So after I posted this I tried to email support for this issue. First reply was to do a repair through Xperia Companion which did not fix the issue. Second reply was "In regards to your inquiry, we appreciate your critics and opinions regarding Sony™ Mobile products. Your ideas are important to us, and as such, we have taken this as a valuable feedback. Also we would like to inform you that you can share your feedback with us through the link mentioned below to our Developer Team for review." which the link is the Sony developer's email address, not a hyperlink. While this may be an issue on the firmware side of things, being I can't find anyone else with this issue I'm not even sure this is an issue outside of my personal phone. Meanwhile, I can't use data saver and as an effort to cut my phone bill, I only have 2GB a month so I'd really love to use it as background data does take quite a chunk out of my data plan with how close I get to 2GB in a month. I don't really want to disable background data altogether as that can also affect use on WiFi which I don't want to restrict.
So I'm a bit stuck at the moment unfortunately.