I used to have a laser engraving business. I would go with the laser engraving, and yes, you do want to put the finish on before having the engraving done.
I would go one step further. Take a photo of the baby and have it laser engraved in the middle of the platter. For this you will want to use plain wood without distracting character (you don't want streaks passing through the image of the baby).
Not all engraving shops are equipped or have the knowledge to engrave photos, you may have to shop around for that.
If you google "laser engraved photos" you will see many examples.