Why can’t lightning tx’s be signed with sighash single to allow for fee updates? Are out-of-band fee payments to confirm a lightning closing transaction the only way to manage a situation where the sat/kw of a previously signed commitment tx are simply below market?

If additional inputs/outputs could be added, fees could be easily adjusted to the fee rate during channel closing and not at the time when the tx was signed...
Let's distinguish a couple cases:

Cooperative close

In the case of a cooperative close, the two channel partner can agree on any transaction they want. Thus, they can pick the fee rate on the spot. As the two outputs would likely not have any timelocks, either party could use child-pays-for-parent (CPFP) to bump the initial closing transaction. They could even issue an RBF transaction initially and update it later if it doesn't confirm quickly enough.

Unilateral close

The party that initiates the closing transaction has their funds locked to allow the counterparty to apply their breach-remedy if an old state was published. This means that traditionally only the counterparty can make use of CPFP to accelerate the closing transaction. This is inconvenient, because there wouldn't be a need for a unilateral close unless the counterparty were currently uncooperative.

The recent lnd 0.7.0 release notes mention that improvements were made to their transaction closing procedures to support both RBF and CPFP.