Your DNS is inside, then if it does resolution only for inside users, you just have to edit the Alias record with the inside IP address, not the public. If your inside DNS serves also outside users, you have to move it to DMZ or Outside and use DNS doctoring feature (Alias command).
I'm sorry but what are the problems you are having? I presume that your internal clients have their DNS setting pointing to the inside DNS server? Do you want your internal clients have their DNS request to be answered by the external DNS server, if so then point your internal clients DNS setting to the outside DNS server IP address (ISP)DNS.
Table of ContentsIntroductionVersion HistoryPossible Future
UpdatesDocuments PurposeNAT Operation in ASA 8.3+ SectionsRule Types
Network Object NATTwice NAT / Manual NATRule Types used per SectionNAT
Types used with Twice NAT / Manual NAT and Network Obje...
Table of Contents Introduction:This document describes details on how
NAT-T works. Background: ESP encrypts all critical information,
encapsulating the entire inner TCP/UDP datagram within an ESP header.
ESP is an IP protocol in the same sense that TCP an...