they don't lose anything if they're not forced to liquidate those bonds but can hold them to maturity. It seems like the real problem here is a lack of diversity in liabilities (all tech/biotech startups).
That's not true. When interest rates go up, they have to pay the higher rates on customer deposits, but they're not getting any more from their bonds. Perhaps they can spread the losses over 10 years, but the losses are the same.
I don't think they "have" to pay the higher interest. Wouldn't they ultimately get that interest from parking money at the Fed over night? A bank isn't going to pay interest on money that results in a net loss, either they get it from the Fed or they are making private loans at a higher rate. Otherwise there would be no incentive to increase deposits as each would result in lost profits.
A bank either pays competitive deposit interest rates, or people move their deposits elsewhere. That can happen quite quickly, and they'd have the same liquidity problem.