It's important to keep in mind that bond funds typically roll over the bonds once they expire, so a short-term, high credit rating bond fund is one of the safest investments around. As mentioned, even if the bonds lose value in the short term due to rates rising, bonds being held are constantly expiring and the notional automatically reinvested at the higher rate. If you hold the fund for about as long as the average duration of the fund, you're very likely to at least break even, even in an extreme rate-rising scenario. And past that point, you're doing well due to the higher rates.
I disagree that banks have a lot of exposure to long term interest rates in the sense that they give you a mortgage - the bank is an intermediary and typically issues its own long term bonds to the market, using the proceeds to fund your mortgage. Then your mortgage essentially pays the coupons of their bonds plus a premium, which they keep. They need to manage the money properly because the interest portion of your payment decreases over time, but it's my understanding that they don't take much risk in general (relatively speaking) due to the many mortgages/bonds they hold/pay at various terms.