hi Tim_Patrick do you have a forwarding (layer 3) virtual server configured? if you do, can you share details of that configuration?
if you do not, then this could be a factor. if you need some additional background on setting up a forwarding virtual server then this is a good reference:
https://packetpushers.net/stateless-routing-f5-ltm/
also, check if your protocol is set to "all protocols" and not just tcp or udp on the forwarding vservers. this will prevent icmp (ping) packets from getting thru if not set to all protocols.
besides using SNAT (auto map is one option, or you can create a SNAT pool), you might also consider leaving SNAT off for this "routed" traffic and create a return route (either static route on the 10.1.216.113 server, with a gateway of 10.1.216.10 for the 10.1.228.0/24 network... or if you have the option to set the route on a router that the 10.1.216.x network is using?)... you could still use SNAT on virtual servers handling specific traffic that you are reverse proxying or load balancing, but for traffic that is "routing" thru the F5 you might find it easier (long term) preserve real IP thru the device.
we are extensively using "inline" F5's where it is either the default gateway (floating self IP as you are) or using the zebos routing with OSPF or BGP to establish the routes, and forwarding virtual servers (without SNAT)... this is on i5800 hardware with 10G and 40G interfaces... acting as a "stateless" router for all traffic, and specific services load balacned / reverse proxied... so the concept you are trying to accomplish is definitely do-able... just may need to ensure your F5 is ready to forward the packets (needs a "listener" aka forwarding virtual server) and routes (direct/connected, or static or dynamic, etc) and then even ICMP will pass thru without issue.
please share more details on your configuration, route tables, etc. and we should be able to get it working!