Who I've just lost a long post but would be really worried in your position.
Have you asked to see their risk assessment for having hatching chicks in class with a dc in there who is severely allergic to all things eggy/chicky?
If not, I would ask to see it, and include as many possible scenarios as you can - from the obvious to the less obvious - things like cross contamination from other dc touching eggs/chicks and not washing hands properly and then touch something your dc touches or eats, need to inform you that there should be more epic pens in school, protocol for if/when your dc suffers anaphylactic shock over and above normal issues e.g. Removing dc from the room away from allergens, decontamination, ensuring teacher not all chick-ified (will other staff need woo pen training?), etc etc
Suspect they won't have thought of many if any issues so it will give you a chance to go in and blast educate and inform them.
Maybe also raise it as a safeguarding issue that they are deliberately and knowingly putting your dc at risk of serious harm or worse by going ahead with this activity. There's no way they would grow a peanut plant in class if anyone had a peanut allergy for example - why the hell are they doing this?
Or threaten to remove him from school while the chicks are there - put in for authorised absence with a reason of 'not wanting my dc to suffer from and potentially endure serious harm or die from anaphylactic shock from a reaction to the chicks in the class as there is a high risk of this' to see what they say...
Either they say fine - bad as it means chicks more important than your dc - so you can then complain to council education officer that your dc is unable to access his education and you don't want to get done for low attendance.
Or they don't grant permission at which point you complain to governors, ofsted, etc that they don't care if they kill your son. Sorry to be very blunt about it but I think you need to use those words to make them realise how serious the effects can be - which it doesn't sound like they do at the moment.
L