Simulating a Reel mower sound

How about recording the real thing?

First off you could buy one for less than $100 new from somewhere with a good return policy, record it and then return it, you wouldn't be able to use it in grass or you won't get your money back but you could at least get the sound of the blades moving.

Option two, around here at least there are several used places that repair mowers and sell used ones. I bet you could pay a small shop like that $5 and borrow one to make a full grass cutting recording.