Is there some aspect of the FMC that inhibits rapid climb?
Absolutely not, if you push the thrust levers to the TOGA detent as required.
In fact, an A320 series aircraft will climb faster & safer than an equivalent 737 in such a situation. The Alpha Protection mode will ensure that the maximum climb rate can be achieved without stalling or overloading the airframe.
This is how Airbus manages such impressive aerial displays and demonstration climb rates at airshows.