Unfortunately, I don't have specific ideas, but some general advice on traveling with infants:

Assuming you don't travel with a nanny, you will spend a LOT of time in your room. In the age bracket you mention, children typically nap twice a day (at least), you change nappies 5 times a day, etc. So unless so you are comfortable doing this at the beach, you will go back to your room each time.

This implies choosing a hotel with minimal distances. Amanzoe (with the beach a drive away) could be difficult, unless you book one of the pavilions right at the beach, or don't go to the beach.

Also, I would recommend booking a "proper" suite, i.e., one with a real door between bedroom and living room. Every child is different, but mine would only sleep at that age when it was pitch dark. If you have a suite, you can put your child to bed, then sit in the living room. This rules out most Amans (unless you book villas). You might be able to sit outside on the terrace, but that's not always ideal depending on weather conditions.

What is helpful in addition is to have a hotel that will provide all the little things that are necessary for children (wipes, nappies, milk powder, baby food, ...). This will allow you to travel a bit lighter. I would say all luxury hotels should do this, but reality can differ. I have had particularly good experiences with Four Seasons.

In terms of dangerous pools, I wouldn't worry for a 4-5 months old. Generally, they can't move at that age. I would worry a bit for an 11 months old, but hotels generally are able to screen in private pools next to the room.

In terms of climate, May-June in Greece would be fine for your child I would believe - but probably too cold for you to swim?

Finally, don't be discouraged. We travelled far too little in the first year after our child was born as we were worried about all kinds of little things. In hindsight, we wished we would have done more.