route problem between SNATed private vlan and hosts on public side vlan

So if I understand you correctly, your setup looks like below.

   Internal           -->           F5     -->      External      &      External
 (192.168.0.10)                                    (10.10.0.0)          (10.20.0.0)

Your current SNAT goes from your internal (192) to your external 10.20.0.0? Do you have SNAT set to automap or a pool? If the .10 subnet is on the F5 and the SNAT is listening for traffic then I'm assuming you're translating it to an address on the 10.20.0.0 network but you don't have a route once it gets there. You could configure a route to 10.10.0.0's default gateway if you wanted to take a round about way to get there.