Disable the 2050 limit in Civ 5 once I've started the game

Normally playing after the end would be accomplished by simply clicking the One... More... Turn... button after the game ends from time. However, this does prevent other victories from being won.

The only other way to change the maximum turns in game would be to edit the save file. This would require a hex editor and a lot more knowledge than I can give you. You could also try using the World Builder to edit your save and turn it into a scenario.

Finally, some quick googling found the Ingame Editor mod. I don't know if it can do what you're trying to do but it looks like it's capable of making a lot of changes in game. It's plausible this could change the max turns (or even the current turn number- that would have that desired effect).

I made a simple web app that will let you change the game parameters of a save file, including max turns: Civilization V save editor

Just open your save file in the app and uncheck Time victory (it will automatically set Max turns to 0). Then download the save file with the changes and open it in the game.

There are a couple alternative options that require either installing the SDK or editing the save file by hand. You can read more details on those here: Is there any way to change victory conditions with the game already started?