I can only speak as a parent, admittedly not of a diabetic child, but they're all the same when they first go away.
Activity centres normally have excellent medical training, in fact now my kids are older, one actually volunteers as a leader at some.
Make the centre aware. Speak to them yourself, and don't rely on the school. Not meaning the school can't be trusted, but that way both you will know what the centre's plan is, (she won't be the first diabetic by any means), and they'll know the personal details with the care.
You're not being over protective, everyone is protective of their kids. But you do have to be careful not to let your daughter think diabetes makes her different and limits her in some way. It means she needs care, but they all need care away from their mums, it'll just mean she grows up checking plans are in place, and it'll help her make her own later.
You'd be amazed how their friends watch out for them as well.
Above all, speak to the centre, and then you'll know what they will be capable of doing, and if they know what to watch out for.