This is easy to solve, and you shouldn't have any problem making this work:
you receive equipment based on a combination of your class and background. Alternatively, you can start with a number of gold pieces based on your class... [table follows] (PHB p.143, "Starting Equipment," emphasis mine.)
At worst, you can just roll for gold and buy, among other things, a shortbow rather than a longbow. Downside: you might not end up with as much stuff as you would have had.
Better (hopeful) solution for you: take the standard starting equipment, sell the longbow, and buy a shortbow off the rack with the proceeds. The standard resale for undamaged weapons is half their list price (PHB p.144, "Selling Treasure"), a longbow lists at 50gp, a shortbow at 25gp, and Bob's your uncle. (A generous GM might let you "trade" your starting longbow for a shortbow and some gold, but maybe not all 25gp of the price difference.)