Welcome to Firewall.cx.
I've been trying to understand the exact problem you have, however the information you've provided is not enough, at least for me.
Am I correct when I say that LAN1 fails to obtain Internet access, where as LAN2 has Internet access without a problem?
If this is true then there are most probably two possibilities as to why this is happening:
1) LAN1 requestes are tunneled through the VPN to LAN2
2) You're missing something in your NAT or ACL statements on ASA (LAN1).
In order to help you, you should post your configurations (Change your public IP addresses) and please provide a clear description of your problem.