There are a number of ways you could experimentally prove that the catalyst does not take part in the chemical reaction.
!. Very carefully weigh the catalyst that you are adding to the decomposition reaction at the beginning and at the end. The weight should not change unless the MnO2 is decomposing also.
2. Check for a change in appearance of the catalyst. If the catalyst decomposes, you should get some manganese metal separating out which would have a different appearance than the MnO2.
3. Carefully measure the amount of peroxide that you are adding to decompose and the amount of oxygen that is released. If you get more than the theoretical amount of O2, then some of the catalyst would have to have decomposed also.