We are currently serving overseas and our storage effects include all the discussed items: We have garden furniture, petrol-powered garden tools (drained and flushed) and all manner of other things. Our Agility contract was locally handled by T H White and they were extremely helpful and nothing got argued about. To be honest, if you are within your capacity allowance, I really can't see what their problem is with non-dangerous stuff like garden chairs. They get paid for every m3 so its in their interest to store as much as possible!
This was all 2 years ago however, so things may have changed...We haven't got the stuff back yet either...