MLS is a bit different than the EPL. MLS is a single entity, meaning the league shares revenue, also player contracts are negotiated by the league. Another big difference is that there is a salary cap of approximately 2 million dollars, therefore you team pay roll has to be 2 million dollars or less. Also it depends on which cup you are talking about, the US open cup (which is similar to the FA Cup) is not run by MLS but rather the US soccer federation. The MLS cup is a playoff format which is after the normal season, the top eight teams go into a tournament and play a series of games (the first being a Home and Home series, the second being a single game, and the third being the MLS Cup final) this tournament is run by the MLS.
I suspect the reason that you don't use money from cup rounds is that the player contracts are owned by MLS and not the specific team...or... or that the cup games you speak of were US Open Cup games and not MLS Cup Games.
I hope that helps you out. To be honest I've only played football manager a few times.