This defines the extension to use to perform the backspoofing. Change the outbound dial to match your inbound DID, terminating on the same Asterisk box as the outbound. The DID must have CNAM lookups enabled. The VOIP provider must allow outbound CID spoofing.
 
The "G" option in the "Dial" command is what does the trick.
 
exten => 1111,1,Playback(enter-phone-number10)
exten => 1111,2,PlayTones(!1400/500)
exten => 1111,3,Read(digits||10)
exten => 1111,4,PlayTones(!1400/500)
exten => 1111,5,Wait(1)
exten => 1111,6,Set(CALLERID(number)=${digits})
exten => 1111,7,Dial(SIP/6304852995@vitel-outbound,,G(default,1111,8))
exten => 1111,8,Goto(10)
exten => 1111,9,HangUp
exten => 1111,10,Playback(that-number)
exten => 1111,11,Wait(1)
exten => 1111,12,ReadFile(CID=/var/spool/asterisk/monitor/callerid.txt,40)
exten => 1111,13,System(/bin/rm -f /var/spool/asterisk/monitor/callerid.txt)
exten => 1111,14,SetVar(LENGTH=$[${LEN(${CID})}-1])
exten => 1111,15,SayAlpha(${CID:0:${LENGTH}})
exten => 1111,16,Wait(1)
exten => 1111,17,PlayTones(!1400/500)
exten => 1111,18,Wait(1)
exten => 1111,19,SayAlpha(${CID:0:${LENGTH}})
exten => 1111,20,HangUp
 
Add this code to the first line of your inbound DID dial plan:
 
exten => 6304852995,2,System(/bin/echo "${CALLERID(name)}" > /var/spool/asterisk/monitor/callerid.txt)