I'm not familiar with the Radioberry specifically, but I've used other SDRs. At first blush, the Radioberry doesn't look all that different from most other inexpensive SDRs, so I'll have a go.
There's no particular reason what you're talking about couldn't work. It looks like that board only outputs 100-150 mW, so even with the preamp board that boosts it to 1.5-3 W, that would require a LOT of gain to do any "broadcasting". How DIY do you want to be?
It would be really inefficient, but I guess you could go the "store-bought" route, and take your pick of the ready-made linear power amps made for QRP HF ham radios, as they'll have the filtering already included. You might even require another amp after that depending on what you want the final power output to be, keeping in mind that you should be be conservative with amateur radio gear that isn't really intended for the continuous duty cycle of AM or long transmit times in general.
I don't want to discourage you as I'm just getting into building transceivers myself, but if you're talking about building an amp with that much gain, it might honestly be less work to build a class D transmitter, and use an audio amplifier board to do PWM on the DC power supply instead of using the Radioberry at all.