If the bracket that holds the sway bar bushing is loose enough, can you push
the bushing sideways to get it out and replace it?  If not, then I would
grind off the bolt head, replace the bushing, and arc-weld the bolt bracket.
It doesn't go through that much stress, so a tack in 1 spot should be
enough.

The car is drivable without the sway bar bushing mount attached but it may
handle a little differently and be noisy.

I have to wonder if your problem is really the CV joints.

A test for CV joints is ---
turn the steering wheel full lock, left or right, and gently apply the gas
as you drive in a circle. Turn the wheel to the other lock and repeat. If
you get a click-click-clidk noise, the CV joints are shot. Each axle has two
CV joints, and inner and an outer. The inner is up at the transmission end,
the outers are out by the tires. The outers wear more, and have a wider
range of operating angles. All CV joints are protected by rubber boots that
will sometimes break open, allowing grease out and water and grime in.
Loading up on water and draining grease out is a huge problem.

I've taken my CV joints apart and cleaned them up and repacked them with new
grease, and had reasonable success. It's been several years -- perhaps a
decade or more -- since I did this last, and the prices of axle shafts has
fallen to the point that replacing them makes more sense than taking them
apart and putting them back together.

Removing the driveshafts is reasonably easy in that if can be done on some
designs without disturbing the alignment -- I would assume that some cars
will have fits with the alignment, but none of the cars I have done this job
on have had issues. I simply replace the dcrive shaft on the affected side,
put the car back together, and it drives as straight as it did before I
started.

Having said that, if you do not have the tell-tale clicking while going in a
tight circle, then odds are good that the CV joints are not the source of
your problem.

You are talking about a stabalizer mount, but describing a motor mount.
Well, depending on where the mount is, I suppose. If you put the car in D,
or R, with the brakes locked, and gently apply the gas, the motor should
remain in place. If it lifts, then you have a motor mount problem.
Obviously, D and R are going to cause the motor to move in different
directions, and this is kind of dangerous so use caution. You don't need so
much gas as to over ride the braking system, but if you are concerned, put
chocks under the tires. I don't know how much life there is in the motor
mounts on your Camry, but I'd not be surprised to find my '88 with them worn
out.

Besides ball joints also check the inner CV or even the outer CV
joint:

http://www.artsautomotive.com/axle_story.htm