I think what you need is to change the ANI based on dialed number that would make all calls to the answering service show your Caller ID instead of the original callers id. Below the NXXNXXXXXX after dnis is the number of the answering service. So you would put something as follows in your outbound voice trunk:
voice trunk T0X
match dnis NXXNXXXXXX ani 4186862300
I only use CLI for my configs but if you look under GUI in Voice -> Trunk Accounts -> Service Provider Trunk -> DNIS:ANI Replacement you can create the rule there. Match DNIS is the number of the answering service ANI is the caller id you want it to show.