Andrew,
To me it sounds like your tippy feeling is very different than my experience. Yes, I have my rear tires filled with fluid. You can youtube it and do it yourself or have your dealer do it which is what I've always asked for at time of purchase. It makes a huge difference even when mowing side-hills. And also, I did recently have my tires air pressure down in the rear (due to the colder whether) and that also made a huge difference. I use a box blade on the back when doing any loader work and you'll be amazed with loaded tires and rear ballast how much more work your loader will do and how much more stable it is.
That said, I'd never raise the loader much on side hills and travel with it as low as possible, but I do a lot of loader work and if on fairly flat ground, I've never had an issue running around quickly with loads. If on a hill, yes, I take it very slow and always have my hand on the joystick ready to drop the bucket to the ground to prevent roll over.
This is my 3rd tractor and the BX's feel much more stable to me than the L's. When mowing, I side hill stuff I never would have considered with my old JD lawn tractor. I just slow down some where appropriate. You'll get the feel of it, but I definitely would get your rear wheels filled asap.
