Hi there, I have a 97 Elise S1
It doesn't say wether you have the Lucas or Cobra system, I have the Cobra system but I imagine both are similar.
I can confirm I get that tone when the car battery gets low (you are right it is the alarm buzzer not to be confused with actuall siren), but you say it has a good battery, I guess this is confirmed by the fact the car starts OK. (If the battery was dead it wouldn't turn the starter.)
However, I also get this tone if the alarm is activated with the door,boot or bonnet open, activated by the push switches, so it's probably worth checking the connections and function of these.
You could try using the fob to disarm the alarm rather than the touch key which may just temporarily disable it.
Probably worth checking the fuse (I think it also powers the interior courtesy light) and You maybe able to test the alarm backup battery by pulling the fuse out (10 amp I think) from the front fuse box. This should trigger the full alarm siren (detecting the main battery voltage loss), I believe the siren is powered by the backup battery.
Handy tip, whenever I am not using the car for long periods, I turn off the immobiliser with the fob and remove the fuse. It leaves the car immobilised but drastically reduces drain on the battery, lasts a lot longer. Just ensure its still charged before replacing fuse and your good to go.
As you said the battery is brand new might also be worth double checking the tightness of the connections.
I guess with the immobiliser bypass you would be able to start the car with the alarm active but I am assuming you have not fitted the bypass. Is it possible the emerald ecu doesn't connect to the immobiliser anyway (not familiar with these) I believe as std the immobiliser (Lucas 5AS Unit also used on Cobra) sends a signal to the ecu via immobiliser module (probably just a relay but never looked) which does the actual immobilising as it where.
The immobiliser module is supposedly mounted in the LHS of the boot just behind the ecu under a riveted on plastic cover.
Hope my ramblings maybe of some help please let us know what you find.