This has been fixed in version 390.147 of the Nvidia driver. I just tested this by downloading and installing libxnvctrl-390xx, nvidia-390xx-utils-390.147-2 from GitLab, and recompiling linux516-nvidia-390xx.147-0.5 against kernel 5.16-rc5 and its headers.
(One thing that looks weird in that process is that you will have to download and extract NVIDIA-Linux-x86_64-390.147-no-compat32.run twice, for both nvidia-390xx-utils-390.147-2, and linux516-nvidia-390xx.147-0.5.)
libxnvctrl-390xx appears to be a new dependency for nvidia-390xx-utils-390.147-2. If there is a conflict with libxnvctrl from a previous driver installation, you will have to remove it. That means answering y to the following question
warning: removing 'libxnvctrl' from target list because it conflicts with 'libxnvctrl-390xx'
:: libxnvctrl-390xx and libxnvctrl are in conflict. Remove libxnvctrl? [y/N]