First of all, it's really not worth trying to play competitively if you cannot access Pokemon that are fully optimised for it -- that means proper EVs, near-perfect IVs and the correct nature. If you don't have such Pokemon, you will be at a constant disadvantage against everyone who does, and in any serious environment, everyone will. You didn't mention what level your Shedinja is, but if it is lacking the correct EVs and IVs, then it is not worth using it in competitive play. If you don't have time to breed and train competitive Pokemon, Pokemon Showdown offers a very good alternative platform that takes all the tedious preparation out of competitive play. It might be in your interest to use that instead of spending time in-game if all you want is to play battles.
To actually answer your question though, it really depends on what perspective you look at it from. Each individual stat on a level 100 Pokemon can gain 63 stat points from EVs and 31 stat points from IVs, but across all their stats, they can gain 127 points from EVs and 186 points from IVs. In this way, EVs are more important for individual stats, and even more important since the stats you'll investment them in are the ones most important to your Pokemon. However, IVs offer more widespread coverage that is, in total, more beneficial to your Pokemon as far as the numbers are concerned. Once again though, it really doesn't matter which is "better", because they are both necessary for competitive play.
Also, you suggested that you would have to breed for both EVs and IVs? This isn't true, you only have to worry about IVs for breeding. All hatched Pokemon start with no EVs, but IVs exist from the beginning and are very difficult to manipulate, unlike EVs. I don't know how to "gen" Pokemon (nor would I mention that since it breaks site rules), but if you're only interested in battles and not in actually owning competitive Pokemon, Pokemon Showdown is a faster and more accessible platform than in-game likely will ever be. (Also note that about six thousand players were recently banned from Sun/Moon for using modified save files on their games, so I would advise against using modification devices now that Nintendo has become serious about removing these players.)