Generally cutting a puffball open is not dangerous. If the puffball is very fresh, as this one appears to be, the interior should be almost pure white, and the spore mass will not yet have matured. As a result, the spores will not disperse when you cut it open.

If the pore mass is mature, it could be anywhere from brown to greenish to purplish - and in that case cutting it open could cause some spores to disperse in plumes. You dont want to get up close and take a whiff of those pores, certainly, but cutting it in half outdoors shouldnt be a problem at all unless you have a particular sensitivity or previous lung problem.

As far as I know there are no acutely toxic spores when inhaled. The primary danger would be getting a large concentrated whiff of spores, which can be dangerous, but which really requires you to get up close and personal and squeeze a puff into your face.

This certainly sounds and looks like a puffball of some kind - the precise texture is the only major question mark, but could easily be consistent with a Lycoperdum species or local species variant.