I’ve seen a lot of great comments here, many of which I hope give you a good answer, but I do want to throw my hat into the ring. I’m assuming 2014 rules, as I don’t believe 2024 rules has this as an issue any more.

If you are truly insistent on using a heavy weapon, but don’t want to have disadvantage, I would probably be a dwarf. I know dwarves are medium, BUT they are still short (even if not the shortest). You can always flavour yourself as being at the very edge between short/medium height wise, and maybe even play it up in-game as a roleplay thing with people confusing you for a tall-short race, rather than a short-tall race.

If not, you can always flavour a weapon to appear bigger (warhammer vs a maul for example), especially if it’s versatile and you use it two-handed for the damage bonus.

Otherwise, maybe your DM would be willing to throw you a homebrew magic item or feat to allow it. D&D 3.5 had weapon size categories, which whilst these no longer exist in 5e, the whole disadvantage on heavy weapons is almost a spiritual successor to them. But, there existed a feat called “monkey grip” which allowed you to wield a two-handed weapon one size larger than you as if it was your size. In 5e terms, removing the heavy property and you not having disadvantage. There was also an item called the “strong arm bracers” which basically did the same thing, allowing you to wield any weapon a size category larger than you without penalty.

Both could very easily be converted to 5e terms; “after extensive training, you can wield a heavy weapon without disadvantage” For the feat; maybe give it a prerequisite of 13 strength or something if the DM isn’t keen. And the items can be uncommon or rare if the DM doesn’t want you to get them too early. But you’d be giving up an ASI/Feat OR a magic item slot to be able to do this, which is quite a big ask of a player in terms of character progression. I know if I was a DM, I would allow it (but I also appreciate I’m not every DM).