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 5W-30 is my drug of choice.
Primary: iPhone 6s Plus 64GB, Galaxy S7 Edge: 5 lines, 3x $5 Android web, 1x $2.99 T-Zones
Secondary: TracFones: iPhone 5s [AT&T] & iPhone 6 [Verizon]
Mobile Broadband: ZTE Z915 LTE hotspot (5GB for $10) + iPad Mini Retina w/ T-Mobile LTE