This is a big "it depends" answer.
You first need to make sure the whole floor assembly is a rated assembly. If it is without upgrades, you should be able to install the fire dampers in the floor if that specific fire damper has been rated by UL to be installed horizontally in a UL rated floor assembly.
My experience with fire dampers is that as long as it is UL rated in the way you want to use it, there are no issues. This may require extra framing around the opening, but that shouldn't be a huge deal.
I really think your biggest issue is the fire rating of the existing floor construction. You may need to add fire rated drywall to get the fire rating you need.