I can't deploy as often as I want because some Sidekiq jobs run for several days, and deploying code means disrupting those jobs

Sounds like a use case for Cadence/Temporal-style fault-oblivious stateful execution with workflows. At last job, we did Unreal Engine deployments with pixel streaming at scale on a huge fleet of GPU servers, and the way we could persist execution state so hassle-free that the code would magically resume at the line right where it got interrupted was so astounding.