Etching ITO on PET without damage
My lab has a lot of experience with ITO-coated PET, but unfortunately I don't think anyone's tried etching it.
We typically etch ITO-coated glass with a mixture of hydrochloric acid/water/nitric acid (4:2:1 by volume) which is fairly fast at room temperature, going through a few µm in about 10 minutes and mask with a positive photoresist.

To your questions:


Will dry film work well for this application?

Because the resolution you need isn't great, it should work fine, so long as the film adheres properly to the ITO.
Underetching is a possibility, but if the features are big enough it should be okay.
The big issue here is trying to use a normal fluorescent light to expose the film.
This is probably hopeless; you're better off finding a proper UV lamp or even using the sun.
How can I determine the thickness of the ITO given its sheet resistance of 50 ohms? 

The mixture I mention above is fairly quick, but either way, I wouldn't worry about the precise film thickness (which may actually vary somewhat depending on the batch/supplier).
The easy way to deal with this is to simply pull the sample out of the etchant occasionally and test the resistance with a multimeter in a region where the ITO should be etched away.
On PET, the conductivity should be negligible when the ITO is gone.
Am I going to destroy the PET during this process?

The datasheet you linked refers to the use of PET containers for storage of chemicals—it may actually last a few hours without significant damage, but I haven't tried it.
Is my plan fundamentally flawed?

I think it's worth a shot.
You will need to handle the ITO sheets fairly delicately—the ITO can crack if the sheet is flexed too much, but I don't see this method as being totally unworkable, though you may have issues with the quality of etching you'll get.


As a final note, I wouldn't worry about getting special glassware.
For measuring simple ratios and etching/developing baths, kitchen quality Pyrex dishes and measuring cups work just fine for this kind of thing.
We keep stacks of them in the cleanroom because they're nice and cheap.