Newbie needs advice, please!

[IslandTractor]

if you have to mow it 2 or 3 times a year then you really have not got rid of it.

If you mow even once a year the multiflora will fade away. You are certainly not going to ever have large MF in those fields. Of course if you stop mowing it will return but that's true in an endemic area if you use chemicals too.

If shoots are toxic for animals then I agree you'd need either chemicals or more frequent mowing.