What type of amplifier do I need for this project?
As I understand it you are using just the horn and not the amplifier that comes with the COGA.
If this is the case, then it won't be driven correctly from the T-shirt amp as (as you guessed) the power output is too low. I would say it's probably a couple of watts at best (as opposed to 40W)  

What you need is either the original COGA amp or something of a similar output rating.
Checking the manual it says the horn is 4 ohms, so most simple amps up to 40W capable of driving 4 ohms will work.
For example, something like this looks suitable.