I bank at a 'local' Mom n Pop bank here in Alaska as well...the last remaining 'local' bank
I've run in to the same issues as you in the past. Specifically with Chrome and/or iCab. It's usually a 'point' release the some how breaks the cookie connection to the bank itself OR...and most likely, it's the bank's software as they've been constantly updating over the past 90 days. I've got a good friend that works for the bank...and apparently they're trying to update their systems for a lot of the new, digital banking and application development for mobile platforms. They released their own little (and extremely basic) app about a month ago. It works fine to check transactions, balances, and other simple options. However...check processing (snapping a picture), transferring between my business acct and personal isn't working...a couple of odds and ends til it's good to go.
You may be going through a similar experience if you're using a 'small' or local M&P bank...they may be updating their software in the background. You may call and check. Others have hit the nail on the head by telling your browser to identify it as an actual browser instead of the mobile version. I use iCab and identify as Safari (OSx). It usually works, but ymmv.
J