To be a pedant I would note that some hacks might change the binary which is in memory, and otherwise might be small enough to fit into a moderately sized cheat list (the DS having reasonably good IF ELSE type loops in its cheat engines, including in nitro hax
http://www.chishm.com/NitroHax/index.html which should allow you to do it for commercial games).
Most Mario hacks I have seen are not that. This sort of thing being more reserved for small changes (that could have big results in game) of a few dozen lines of code at best (your average level hack being many times that, music even more so, graphics... oh dear).
More proof of concept still is maybe not with a DSi but with a GBA if you loaded up the data you wanted to the GBA slot, booted to DS mode you could probably alter the game code with enough cheats (which you can do with a retail cart) to go speak to the GBA slot and fetch data from there.
I am not aware of any services that might program a replacement ROM chip similar to what we see on the GB/GBC where people might take certain games, redo the ROM chips and have a flash cart for their trouble. Likewise you are going to be hard pressed to find a repro option.
Further to the GBA slot stuff if the game is old enough or you are bored (and talented) enough to hack it suitably then you could possibly flash your DS firmware, use a simple GBA flash cart as a DS flash can and autoboot the modified DS game on that but still have the save written to the baseline DS game (technically would be easier than making the save work on the GBA slot).
All this is utterly obviated though by "get a flash cart"/modify your DSi or 3ds, and if you somehow need the hacked version's save on an original game then writing such a thing will take longer* to boot the program and select the file you want written.
*give or take maybe Warioware DIY and Daigasso with their stupidly large saves, which might also work as a secondary injection point/data source for those games and maybe in the case of some more conventional games if you are willing to sacrifice some save data too. I won't speculate on making a save game exploit with your hack in it but it is within the realm of reason).