So I've been a busy beaver, I've changed every suspension bush in the car, changed the front and rear ball joints and fitted coilovers.  Next up I'm fitting new adjustable swaybars.  The front went in fine but I can't figure out how to install the rear swaybar.  Does the rear cross member need to come out?  If so, can I just loosen it and drop it down a little?

You can loosen it enough to get at the bolts that hold the sway bar brackets in place

TomsMR2
You can loosen it enough to get at the bolts that hold the sway bar brackets in place

So I presume I support the car body on axle stands then support the cross member with a jack and loosen the four bolts that hold up the cross member?  Then I just lower the jack a bit to let the cross member drop and I can get to the bolts? 

^ yes..
also take the thru bolt out of the engine mount on the crossmember

Slightly off topic, where did you get your ball joints from? Have been looking around and can't tell what my best option is.

I just got them from eBay:
Front Ball Joints:  https://www.ebay.com.au/itm/Front-Lower-Ball-Joints-FOR-Toyota-MR2-AW11R-SW20R-Tercel-AL25R-08-1986-10-1/251804208834?hash=item3aa0b34ec2:g:m~AAAOSwkl9dXQyt
Rear Ball Joints:  https://www.ebay.com.au/itm/REAR-Ball-Joints-SW20R-AW11R-FOR-TOYOTA-MR2-1986-1999-SUPER-TOUGH/253077281569?hash=item3aec94db21:g:4DkAAOSw4QRdyY8f

No problems so far?

No, they were a perfect fit.
 
The suspension on my car appears to have been untouched for 260,000kms so it was all pretty average.  I think they were original ball joints and they took a fair bit of effort removing.  I normally don't need a ball joint removal tool, I just belt the control arm with a hammer and they fall out.  These were really tough to remove, I ended up using one of these: https://www.toolmart.com.au/sp-sp67023.html