That's interesting. What is that website? Seems like you have a good grasp of the fact that it's not a phone setting issue. One recommendation is a VPN which will hide your traffic so I assume they won't be able to block you. I use Private Internet Access w/ their annual plan. Don't really use it on my phones/tablets but I do have it set to automatically sign in on my computers.
Castrol EDGE 0W-20 is my drug of choice.
Primary: iPhone 7 Plus, Galaxy Note 5, LG G5: 5 lines, 3x $5 Android web, 1x $2.99 T-Zones
Secondary: TracFones: iPhone SE [AT&T] & iPhone 6s [Verizon]
Mobile Broadband: ZTE Z915 LTE hotspot (5GB for $10) + iPad Mini Retina w/ T-Mobile LTE