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: Galaxy Note 3 and LG F6: 5 lines, 1x $15 5GB data, 2x $5 Android web, 1x $2.99 T-Zones, corp discount + $10/mo loyalty discount = $93/mo tax inclusive
Secondary: iPhone 4 & Optimus Dynamic II: TracFone (Verizon); Galaxy Epic: RingPlus (Sprint)
Tablet: iPad Mini Retina w/ T-Mobile 4G LTE