If this is happening due to external DNS resolution, use the "dns" keyword of the static command. (Pix 6.2+)
static (inside,outside) 200.1.1.1 192.168.1.1 dns netmask 255.255.255.255
This tells the Pix to "doctor" the DNS reply and substitue the private IP address for the public when the DNS response is returned.
If anything below 6.2, you can use the "alias" command to accomplish the same thing:
static (inside,outside) 200.1.1.1 192.168.1.1 dns netmask 255.255.255.255
alias (inside) 192.168.1.1 200.1.1.1 255.255.255.255
Note that the Pix's PDM does support the "alias" command except for use of the Monitoring tab.
If the probem is occurring due to internal DNS resolution, you must fix it there. You cannot access a resource on the inside from the inside by using the Pix. The Pix does not allow a packet to enter and exit the same interface.
If the problem is due a DMZ configuration on the Pix, you can use bi-directional NAT. (Pix 6.2+) If this is the case, I'd be happy to give examples for this too.