Saucepan Accessory

Using a drop-lid or otoshibuta is commonly used for Japanese simmered dishes. Take a look at the lids shown on this site: the metal ones are more or less the shape you described, albeit without a lip. Although Japanese drop-lids typically sit above all the food and liquid, I see no reason why they could not be slightly submerged. Even sitting on top of the whole stew, a drop lid would significantly decrease evaporation.

The French tradition uses a piece of parchment paper instead. Like a drop-lid, it simply sits on top of the braise and prevents evaporation similarly. This article provides instructions for using one.

It sounds like you're looking for something like a steamer basket:

These are normally put in the bottom of a pot to hold food out of the water so it can steam. The flaps flip out to fit pots of varying sizes. You could possibly instead use it on top of your food to hold it down into the liquid.

Similarly, the are many strainer baskets that come in varying sizes that can fit inside a pot. These come in a variety of styles, from a traditional colander to a "pasta insert" to hold pasta while it cooks for easy removal. If you can find one to fit your pot, it may be another excellent option to put on top of your food to hold it down.

You might also find a round cooling rack that fits inside your pot to use similarly, and use it to push the food down under the liquid while it cooks.