Ok, interesting idear, there.
But I R not your expert -- even though I did own / operate a Trust company for a few years. (Those were some fun, fast days, btw.)
I think you have a model of what you are describing in the Savings and Loans "crisis" of the early the 1980's. They had long-term -- often mortgage based -- low interest rate income, and then, as you described, had to compete for deposits into the double digits, when the Fed changed the game.
Since under Reagonomics and the Fed we were not going to deal with the real problem (the US had become an energy importer and we were beginning to sink under the debt load), and instead were going to fudge with the money, the money itself was becoming a toxic problem. The brainless mantra chant of the day (which continued up to the crash two years ago) was De-Regulation! De-Regulation! So that was the claimed fix for the Savings and Loans.
The S&L's were "de-regulated" to go into very risky and insane commercial real estate deals, where properties were "flipped" and prices ran up to provide false profits. (sound like a recent game with the commercial banks?) The old time, moral and decent directors of S&L's were removed from the boards and positions with the older S&Ls so that new "smarter" operators could come in and totally load them with bad debts, and asset strip what was left.
By the late 80's most S&Ls were wiped out shells and the US had found its way into Debt Addiction.
Banks, at the time, were not so much into long-term mortgages, so they did not tend to be trapped in the interest game as severely as the S&Ls were.