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 6 64GB, Galaxy Note 3, Lumia 635: 5 lines, 3x $5 Android web, 1x $2.99 T-Zones, corp discount + $10/mo loyalty discount = $93/mo tax inclusive
Secondary: iPhone 4S & LG Ultimate 2: TracFone (Verizon)
Mobile Broadband: HP Stream 13 w/ T-Mobile 4G & iPad Mini Retina w/ T-Mobile 4G LTE ($10/mo for 5GB with hotspot)