My dealer told me the same thing, they also told me the wheels could be flipped. These small tractors are not the bread an butter of the dealer. I would trust them to fix it if something went wrong but I don't think the service dept does a lot of work on them otherwise.
I also take it with a grain of salt when someone says something like "We've done it hundreds of times".... do you really think they've sold hundreds of these tractors?
I'm sure that, like I said, if you got the chains on tight and they stay tight there is probably little danger. There is enough clearance for the chains, just not a lot of extra room. If the chains are loose they will likely hit the bracket and what is on that bracket. Eventually, even if nothing catastrophic happens, loose chains would slowly and eventually cause damage that needs repairing. If you put chains on without the spacers you need to watch them closely, check them often, and tighten them up if they get loose. A set of chain tensioners would also be a good idea.