This sounds like an integration issue: systems like LaunchDarkly typically allow you to specify a default value for when the feature flag server can’t be reached.
And/or build a near cache so you treat the 3rd party as a control layer, but actually serve requests from your near cache as data layer. Then when 3rd party goes down, your app doesn’t notice at all, and you can still manually update/override values in the cache in emergencies.