The issue is really with the consumption plan of Azure Function when it comes to the "cold start phenomenon". In this case you can try to implement a keep-alive mechanism to periodically invoke the function, be it a timed trigger or HTTP trigger which will ping your Service Bus Function.

OR

you can switch to flex-consumption tier which supports always ready instances feature: https://learn.microsoft.com/en-us/azure/azure-functions/flex-consumption-plan#always-ready-instances

Hopefully that helps !