This sounds like an immobiliser fault NOT an alarm fault. The alarm does nothing except beep at you.
The immobiliser is the EWS. Unless you have a suitable version you can’t remove it and even then it’s a by-pass procedure involving re-programming the DME. You don’t just take out the EWS. It just isn’t going to start without it.
The EWS looks at a number of things and decides whether it’s happy to release the starter. If you get no crank and you are happy it’s not your battery it’s likely to be the EWS.
If you get a crank and no start and the battery is dodgy or has just gone flat it’s likely to be the next process up the chain which is the EWS handshake with the DME which can get out of sync. That needs a rolling code reset with an INPA laptop. I’d rather have that issue to be honest as it’s easier to resolve when you know how.
Back to your issue...
We’re on holiday right now so I have no access to my detailed notes.
But by way of example the EWS would want to know that a valid key has been detected. The key has a transponder in it. That talks to a ring in the key barrel of the steering column. That check has nothing to do with the physical fit of the key and does not require the battery in the key to be working (that’s for remote unlocking only).
The other conditions I can’t remeber off the top of my head but I would expect to include that the car is not locked. And if it’s automatic that the selector is in neutral. For US models it also requires a foot on the brake pedal. Common issues here are faulty switches that detect the pedal position.
As a first step the EWS needs to be replaced. And you need to make sure your battery is charged.
You need to get yourself a code reader and report back. The codes may or may not help in this instance. But normally reading the codes is always the right advice. And a £50 investment for a decent tool is an excellent and frankly necessary purchase for a Z3 owner.
But ultimately it may require a multimeter put on the connector block of the EWS to check the hi/low state of each circuit corresponding to the various checks. It’s easy enough with a pinout and logic table. Both of which I have at home.
If all these checks pass then you need to see whether the EWS, which is really just an intelligent relay, is providing output to the starter circuit. If it’s not it will be an internal fault, ie the relay mechanism itself is broken or stuck. Which is exactly what was wrong with mine and caused many, many hours of research to understand this system!
One more note of caution. You have a very early car and there are a few different versions of the EWS. In all honesty my research did not go back to the earliest cars. So if the advice above seems illogical at any point it may be that you have something more primitive fitted.
While you’re waiting for your code reader, you could check that the starter motor spins by hot wiring. Again I don’t have my notes at hand but I’m sure help will be along here to advise or just google it and you’ll find it easy enough.
That’s about as much as I can remember off the top of my head!
