Skimming the manual, I wonder if this wouldn't work if you configure the entry system to operate in call forwarding mode, specifying a 3CX extension number as the number to dial, then connecting the telco connections of the unit to an analogue adaptor.
That would, effectively, remove it completely from the existing phone line, and you might be able to do it without the bypass module too, which seems to be there largely to allow the gate system to send ring tone to the house without upsetting the telephone exchange.
I suppose you could connect in normal mode, to the 'house' ports of the gate, depending on whether you have an FXO or FXS adaptor, and the name you give the adaptor in 3CX will provide the ID on the house phones, rather than a distinctive ring.
Either way, I would have thought that the bypass model is probably superfluous, if you're having a dedicated connection between the gate and the PBX.
I have a similar entry system, which I have configured to dial a ring group on my 3CX, though it's designed to connect directly to a line, without a bypass module. In my case, it's set up to dial 81# - 81 is my ring group for the entry phone, and the # just prods the analogue adaptor to complete the call faster.
A brief note about it is at http://gonedigital.net/2010/04/20/making-an-entrance/