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I'd also go with making the peg smaller rather than the hole bigger. A tool like this one (and there are others as well, of course.)
Or, maybe, you can do it with the DIY version like this:
A positive attitude may not solve all your problems, but it will annoy enough people to make it worth the effort. ~Herm Albright
Most assuredly don't ream out the peg box. That could spell the end of your instrument as you know it. If you weaken the wood fiber, the peg hole could crack just from string tension. The wood itself could split just turning the reamer.
Go with the pegs. As you reduce them, either with a peg cutter or sand paper, periodically insert them until they seat firmly and twist. Then pull out, you will see where the pegs rub. You'll be able to determine how much more to take off.
Insure you set the pegs on top of the box and measure all 4 pegs before you start. You'll be more likely to get them all the same uniform length.
"I find your lack of Fiddle, disturbing" - Darth Vader