Greetings:
I am assuming that you also opened the mortise with an auger of some kind but have failed to mention it. This operation should deal with the depth of the joint. There are certainly many out there who can help you find a muscle driven machine.
Regarding round pegs: many folks use them but the sawn or riven octagonal is also popular. Is the square peg truly square or just the part you can see?
Back to you.
Curtis