iPhone shuts off when I use the internet on the 3G network & I can't upgrade/restore because of network connection error. Any Help?

A few months ago, while on the 3G network, my iPhone 3GS starting turning off randomly when I was using safari or an online app and the battery was somewhere around 5-20%. When this happened the only way to get the phone back on was by fully charging it which sometimes took a few hours even. It wasn't a big deal at the time because it was infrequent and while the phone was at a low battery %. However, in the past few months, around the time I upgraded to iOS 5.1 it started happened more and more often and at higher and higher battery percentages. It is now at a point where my phone shuts off every time I connect to the 3G network regardless of what my battery life is. I have called Rogers several times to complain about this and I have been told it is because of the battery for the most part but I don’t buy it; the problem has got to be more significant than the battery alone because I can listen to music and send texts at anytime and fully use the internet when I am connected to Wi-Fi. I more recently talked to someone in tech support from Rogers who said my phone cannot support the iOS 5.1 software and that is why it is crashing. They also asked if my phone was heating up more now than in the past and it is. They said this was further evidence that my phone was couldn't handle the new software and was basically on overload. I was told to the best solution was to "Back Up" my iPhone then do a full "Restore". Here is where my next problem comes along. I cannot "Restore" my iPhone without doing an upgrade and every time I attempt to "Upgrade & Restore" an error message pops up that says "There was a problem downloading the software for the iPhone "...". The network connection timed out/was reset. -Make sure your network settings are correct and your network connection is active, or try again later". I have followed all of the protocol that is readily available online to address this problem but still no success. I disabled my firewall and anti-virus programs, I have ensured my connection is active and stable, I have upgraded to the newest iTunes, I have uninstalled iTunes and all apple programs in the correct order then reinstalled iTunes, I have disabled my DNS cache. Nothing works.

If you have any advise whatsoever please share as it would be greatly appreciated.

Thanks