I don't know if they are sold in the US (not even knapsack brushcutters are sold anymore apparently), but you can use a carbide-tipped circular saw on a brushcutter. Makes short work of pretty much anything and can be simply cleaned with gasoline, diesel fuel, brake cleaner etc. I have one and it can but pretty much everything.
Typical procedure to get rid of ivy (especially English) in New Zealand is to have one operator with a carbide saw clearing a path in the ivy, followed by another operator about 20 yards behind with a power sprayer loaded with glyphosate or another post-emergence herbicide. As long as wounds in the ivy are open, the herbicide will get in the vascular system and kill it. It does work well to get rid of very serious infestations, but it's expensive and/or time consuming.
Another invasive plant that gets everywehere here courtesy of birds (and rabbits) is Deadly nightshade (belladonna). Technically it's not invasive because it's native, but it really gets everywhere. Like Kat says people don't know what it is and let it grow, not exactly responsible behavior if there are children running around since the berries are incredibly toxic (unless you are a bird, a conny or a cow
) and very attractive. Luckily it's easily pulled from the ground and killed but them birds really spread it everywhere.