Load-Balancing Client/Server on same subnet

You do right, your issue is on other side (server side).

If you do not use SNAT, IP address used to open the server connection is the client IP address (and the mac used is BIGIP's one). If you use SNAT, the IP address used is the BIGIP's selfIP (and mac as well).

In your case, servers see client IP address in the connection as source but in the same subnet (broadcast domain). Switches as well (and one switch knows your client MAC).

I would say, if one server sees ARP request / response / broadcast from the client 10.1.1.12, the server should learn this entry and so, send traffic back to the client instead of the BIGIP.

Take a trace and check ARP / IP layers. You should see ARP requests for 10.1.1.12. And should see a difference between working and non working TCP connection at layer 2. I do not understand why working servers don't have any entry for 10.1.1.12 if the original TCP connection uses it.