I don't know much about the clumping varieties (which I hope it is) but I've had a problem with being invaded by running bamboo for years. IT was enough to cause me to abandon gardening in that area entirely as, the more water and fertilizer I put on MY yard, the more I encouraged the wretched think to intrude and sucker in my place.
I've a couple of suggestions and
First is to try to identify your monster, with some photos, distance/ close up, clump bases, canes, leaves etc and maybe some physical samples if you can get them. You can possibly do the identification on-line but if not there are a couple of bamboo specialty nurseries who might be able to help you. I say that because they are all anxious to push the benefits of the clumping varieties and NOT have them classed with the weedier ones. I know there is a bamboo nursery hinterland round oxenford way, and a couple further north Qld. If it turns out to be a "weed" you may have better prospects.
Second armed with at least a proper botanical name you can do some more research on what the plant's growth is and the long-term prospects
Third , make yourself familiar with the Qld neighbourhood disputes legislation - it's probably early days yet and I always urge neighbours to try to resolve differences themselves, a bit of give and take armed with information and a list of complaints and a list of possible cures works better than the animosity of a paper war.
It is my understanding that bamboo are pretty shallow rooted so the matt-like fibrous mass you found is probably it. If that alone is a problem with your plantings, really the only way is to get some professional help to trench along the boundary to a decent depth (there are companies that specialise in this with water blasting, cuts through like a knife) then with a reasonably impermiable barrier to stop the roots from further invading. You'd probably have to go down a metre. Then, if its not a running bamboo, the cut-off roots would simply die but it would be better to just dig over the ground there and pull them out.
I also believe that cut stems do not re-shoot. It's really only a super tall grass.
The material itself, probably would not compost down easily, it's been suggested using the stuff as an alternative to wood pulp for paper-making, certainly the canes themselves will last for some years. So the mess as well as the shade etc is another point to add to your list of complaints.
If you have no invading suckers but the root problem and shade (plus falling debris) them you are largely restricted to using shallow-rooted plants yourself, or potted ones. Or using a weed-mat type material (or even denser) and raising up some garden-beds over that, to take shade plants. there are lots of alternatives there, bromeliads, orchids, azalias etc.

Here is a link that might be useful: the Qld Justice Dept website

g'day DD,

really would need to know is it a native clumping bamboo? they are not supposed to be overly intrusive eg.,. like spreading rapidly, as for them growing tall and dumping lots of material all the ones i see do that, so this might come back to you talking to the neighbour to cut them back and clean up.

might be redical but maybe you could help whether labour or money, just looking for an amicable solution.

legally all you can do is cut overhangers back to the boundary line (nice and neat of course) and place those cut offs back over the fence same with the litter, that's the law on the subject.

so might be time to sit with neighbour and share a drink and a chat.

if they are wildly spreading that can bring in other issues, like interfering with your paths and foundations.

so talk first.

len

Here is a link that might be useful: lens garden page

The roots will continue to grow horizontally no matter how many canes or leaves one cuts. In fact a path can be made in the middle of a Bamboo Grove! Each Grove can be easily structured, trained. The height is stopped when the top of a cane is cut. It will grow no higher, ever. If side leaves are cut they will not regrow. However, the Grove (Plant) itself will continue to expand, grow horizontally. To control that there are two ways. Cut the roots, underground, by 18" deep and remove a clump. Each year. The other option is to use 80mil plastic to stop the Grove from expanding. Dig down 24" and place the plastic vertically. Black Bamboo is beautiful and very easy to control in height and surface width. The leaves are a natural mulch for garden soil. Grow vegetable gardens in and around Bamboo Groves in Pots.

Depends on which bamboo it is, there are hundreds of different ones.

I had running bamboo on my place in the Blue Mountains, west of Sydney. The council contacted me (I was renting the place out) and told me to get rid of it. I paid someone to come in and poison it. I dont know what your council's attitude is to this kind of pest. Currently (in the Shoalhaven) I have a neighbor whose place is overrun with Morning Glory, and I have to diligently patrol the boundary to keep it from infesting my place. I also would like to solve this problem, with out angering the neighbor of possible.