I generally make mine with tapered rods and holes. 3/8" upper tapered to 5/16" at the tip. I made myself a square tapered reamer out of a piece of 3/8" round stock. Not a fun project, but it does work surprisingly well. Anyway...
For something that you don't want to put too much effort into, and you're determined to use a larger rod, you could make yourself a scraping tool. I have one I use to help correct errant undersized rod holes. A piece of 1/4" steel rod long enough (48") to hang onto and work with, and peen the end just a bit to get yourself a "hook" to scrape with. If you do it right, you can cut surprisingly quickly with it. Just make sure you have plenty of room in whatever direction you want to go, clearancing the lock bolt and staying away from the bottom of the stock, etc. Easier to enlarge the hole "up" towards the barrel than "down", just because there ain't an easy way to work the tool and put pressure on it to scrape "down".
To drill it out, without buying a purpose-made drill bit, you can take a regular drill bit and braze it to the end of a steel extension rod. I did one once with a screw tip spiral auger and threaded the lower end of it, and then put matching threads into the end of the steel rod to extend it. It's not exactly a precision instrument, but it works, more or less. Now to open up an existing hole you can't use an auger, but will have to use a standard twist drill bit.
OR....
You could leave it be, since the 5/16" rod will be fine. :wink: