I know you want small but I had a Homedepot one on my basement door for years. Not the smallest and prettiest but loud and only cost around $15 each
What requirements does your municipality have for safety systems for entry to the pool area? Be sure to check that.
Just keep in mind it is highly unlikely that any of these door chime systems meet code. In almost all locales it requires either a barrier or an alarm. I know it’s common practice to remove the alarms as they are annoying but these little magnetic contact switches are not a replacement.
They will definitely have swim lessons! Since we will have the pool accessible from the house I want multiple layers of safety. Swim lessons will be the final layer if everything else fails.
On a related note, are there any water disturbance sensors that are worth looking into?
You really need to look at what needs to be done to close out your pool permit. I have the same setup as you (entire property is fenced and house is used as a barrier). I need to have EVERY door that leads to the back yard alarmed and sound for at least 30-seconds and also an alarm in the pool to detect entry into the water. If I install an automatic pool cover, the previously mentioned items are not needed.
Good luck. A quick call to your building or permit department should be able to give your the requirements.
Edit - you may want to look into your fence gates to the back yard as well. Mine need to open out from the backyard and be self closing. Assuming your town has similar requirments.