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I destroyed a back tire on my 2008 Subaru Impreza manual transmission with viscous coupling.

The manual says all four tires must be the same size.
The spare tire must only be mounted onto the rear axle when applicable.
I don't need to pull a fuse to disable AWB on my manual, but automatic transmissions do.

I talked to a tire mechanic who has been doing this for decades. He says that I do NOT have to replace all four tires with new ones. He said replacing just the single blown rear tire with a new one (same general size, but different brand and likely slightly different circumference) will be totally fine. He says he has never had any problems doing it on other Subarus. In fact, the spare tire that I was running on WAS of a different size and it ran on it just fine for 75 miles and no warning lights.

In this case, is it ok to drive around with one new tire on my Subaru? Or do I still have to replace all four?

The mechanic said that as long as the tires are all of the same size (205/55/16) little differences in tread depth and manufacturer differences won't matter.

Driving around with tires of different rolling radii will cause the viscous coupling to get hot / overheat and eventually fail...

As it will probably happen long enough after the tire fitter did the tire, then he won't have to pay, but you will...

This is a known issue, both on Subaru and Volvo car with awd and viscous couplings, however, you have read the manufacturer's instructions, now it is up to you.

Just out of interest, here is a similar answer: https://mechanics.stackexchange.com/a/51301/10976

It is possible to get new tires shaved down to the same tread depth as the existing worn ones. It may be worth getting that done, then you wont have to replace them all.

For example this article says -

Buying four new tires may be needlessly expensive for drivers who only
need a single tire to join the three other moderately worn tires. But
you can buy a single new tire from $BIG_TIRE_VENDOR and have the
company shave it to the tread depth that matches the depth of your
other tires. It will shave any tire you buy from the company, usually
for $25 to $30.

Note that shaving a tire will likely nullify its tread-wear warranty. Other retailers may offer a similar service, though the special equipment to shave a tire’s tread isn’t common.

So, no, it is not absolutely necessary to replace all four wheels.

Subaru's have lots of quirks, and tire size is one of them. Talk to a Subaru mechanic rather than a tire-changer, and you'll get a completely different answer.

Subaru's are well known for having tight tolerances on wheels. If you cannot change all four at the same time, buy a new tire of exactly the same make and model, and have the new tire shaved down to the same size as the others.

The tolerance on Subarus is 1/4" in circumference - which translates to between 1/32 and 2/32 of an inch difference in tread depth (or radius).

You might think that shaving a tire to be smaller is an expensive waste of money. But think of the cost of replacing your center differential (well into four figures - often cheaper to scrap the car), and it's cheap insurance.

You don't need to change all four wheels when one gets destroyed. For an emargency run you can use any wheel that fits to the hub. For a regular use I would consider the effects of mismatching tyres on the car reliability and behaviour. You can easily find that saving a buck may lead to a hundred-buck repair elsewhere.

Changing all four wheels for the very same brand, model and dimensions means that all four wheels will behave equally. The ABS wont detect different wheel speeds (no false positives detecting wheelspin/wheelslip) and viscocouples will be loaded only when their opperation os needed.

If you compare tyre radii and they match the couplers and electronics will be fine. The bigger differences in wheel dimenions, the more unnecessary load on the viscocouples and more "thinking" of ABS unit.

Different tyre models and brands behave differently and the harsher conditions the bigger the difference. Try to have same tyres on all four wheels. If not possible (financially acceptable) try to have same pairs on front and rear wheels.

Trust me, you don't want a car to twist during emergency braking or perform any other unexpected move.