How do you have the trunks set-up? Is 3CX registering to the Alcatel as if the Alcatel were a provider, using a generic trunk? Is the Alcatel registering to 3Cx with 3CX set-up as a slave Bridge trunk? Is a trunk from one end registering to an extension on the other PBX? There are a number of ways of interconnecting, each will send caller ID in different ways.
Even between two 3CX PBXs using a bridge trunk, caller ID does not behave the same as within a single PBX.