What is the best way to keep the SSH agent responding to background requests?

Good to clarify: the SSH private key is still tied to the 1Password "lock state", so locking 1Password means that there's no more private key available until you unlock, even if the 24h setting is used. The 24h setting will only persist the fact that you approved an app. But the first SSH request you make after 1Password locks would still require a regular unlock, "resuming" all previous approvals from the last 24h with a single unlock prompt.

I also get repeatedly asked to unlock 1Password for no apparent reason; I suspect it's because of something SSH-related going on in the background.

Also good to note: the regular unlock prompt does not support background suppressing yet, so for better autofetch behavior, you may want to consider switching the "for 24 hours" setting back to "until 1Password locks". This way, you'll always get the SSH authorization prompt, which supports background suppressing and better attribution to the source of the request.

We're looking into also adding similar behavior to the regular unlock prompt for a better experience around Git autofetch + the 24h setting.