Just to go over old ground,yours is a Lag 1 Phase 2,and these have a key transponder chip and a reciever aerial around the ignition barrel,and not,as you quite rightly say,the immobiliser which works off the central locking remote blip.
In theory,it's either the key transponder(unlikely,as neither key seems to work),an intermittant fault with the aerial or wiring to it,or the
I don't know of any common faults with any of it,to be honest,so if the cost of having the
modified isn't too prohibitive,I would have thought this would be the way to go.Just for safety's sake,I'd then have an aftermarket immobiliser fitted.If you have a Thatcham approved one fitted by a registered company,then you might even find that your insurance premium goes down fractionally,although you must tell them the manufacturers immobiliser has been diasbled.
Hope this helps,if not,others may have better ideas.