Is there a way to upgrade my existing ships?

I just completed the research for Coilguns and would like to upgrade my existing fleet for a damage boost. However, I'm not seeing an option for this anywhere. The only avenue I'm seeing is to design a new ship that uses Coilguns and build some of those. But if I have to keep incrementally upgrading tech and buying all new ships, then doing that again when I research the next level of tech, that seems horribly inefficient and I might want to simply wait for a few techs and batch them up in the new ship design.

However, if there is a way to partially upgrade already existing ships, that would be a huge benefit. I see an "Upgrade Fleet" option, but it is greyed out.

So to upgrade your fleet you first need to update the ship design, you can go into the ship designer in the top left and add the Coilguns. What I think you may have done wrong is created a new ship design and added the Coilguns there instead of updating your current Papillon-class Corvettes. When you change your ship design make sure you are modifying the Papillon-Class corvette and not creating a new class.

If you want to change the name of the class you can do that in the ship building screen, if you create a new class it won't change the existing class. Once you have update the Papillon-class and/or renamed it the upgrade fleet button should be Colored.

Right click in your home world with the ship selected and there will be a message to upgrade the fleet to the latest designs.