The mower can cut clean, thin bark without harm but it should not be asked to cut thick bark, let alone woodchips. It especially must not cut sand or grit. If you look at the stuff that you call tea tree mulch by the time it gets whippersnipped and thrown across the lawn, you'll probably find it contains lots of sand and grit, and very likely some woodchips as well. There might even be a few paperclips, kids' toys and loose mower-parts included.
I suggest you look closely at it, then think about whether it is suitable for mowing.
