Persistence which doesn't bypass the LTM

Can you clarify what you mean by 'sessions bypassing the load balancer'? Based on face value this generally indicates a routing issue in the rest of the network infrastructure. (Or a server responding with hard-coded links to itself in some cases...)

'Upstream' routers will need to route F5 objects (VIPs for various listeners) to the LTM. 'Downstream' routers will need to route return traffic back to the LTM. If your LTM is not in-line with the default routing pathway, you may need to set up features such as SNAT or policy-based routing to ensure return traffic traverses the LTM.

Persistence is used by LTM to ensure connections are sent to the same pool member based on a trait (cookie, source IP, etc.). It does not impact routing outside of the device.