After doing some googling, I found a number of different pronunciations:
Shy-rus
Sheer-us (this is the one I've always heard)
Share-us

This site has a number of possibilities. There are 4 different English possibilities depending on which audio button you push.
SHIRAS
Is there one single correct one?

I ran across this pic of a staged moose shoot in Norway. Pretty funny. That's got to be saddest 'moose' I've ever seen.

Damn, that looks like Pickett's charge at Gettysburg.

Saw a nice one Sunday morning while climbing one of the 14er peaks preparing for elk season. Solid black, trophy shovels.

A moonbeam hiker that I chatted with on the trail told me that she had 'sensed the presence' of a moose. I could tell that she didn't mean 'sensed' it with one of her 5 senses like a normal person.
"'Happy Harvest Moon' to you too."

It's a long "I" in Alaska.

The Black Phase Brown bears of the ABC islands are all referred to as Shirus Brown bears. At least when I was guiding there. I had a client shoot one on Kuiu island which has no brown bears. Thought it was the biggest black bear in history when I saw it. How did it get here? I dunno, guess it crossed at low tide between islands?

I radioed the state patrol shortly after realizing the mistake. Black color Brown bear on an island without brown bears! Jet black as coal but with a dull colored hide, not the typical shiny black coat of a black bear. However when they are in the water and all wet, you cannot see them very clearly anyway. Well except it was a giant!

F&G took the bear hide I skinned for them( trying to be as helpful and polite as possible). The F&G guy on the plane said, I would have shot that one in a heart beat on this island too! No trouble, glad we called it on and did not try to do something stupid! We carried on with the hunt for a black bear. After they left. It was about an inch shy of 9 foot. Best I could measure on the beach with a 6' fabric tape. Might have been a bit bigger...... or smaller?

It's got a better Twang to it then saying it's a black brown bear. I've also only heard this in Wyoming when referring to the moose there. Shirus Moose. They are also quite dark, or near black in color.

Shy-rus is the only pronunciation I have heard. As a student of Etymology it made me look for it and I find nothing else...

This Shiras was up in the driveway at the ranch (Colorado) a few weeks ago when I had to make a run up there.

[bleep]' moose!

Uh, bleep. Yeah,that's what I meant to say.

Me too.

Sheer-us is the way I've always thought it was pronounced!

Me 3

http://www.pronouncekiwi.com/Shiras%20moose

FWIW.

That's the link I had in my OP. They have 4 different pronunciations so I guess it's 'take your pick'.

Sorry. Didn't click the link, so missed it.

I'm in the Sheer-us camp.

Since Idaho only has 1 subspecies, the usual pronunciation is 'moose'.

That link is pronouncing it as though there is an apostrophe before the s, as in Shira (a noun) owns a moose. An old book (1892) I have here lists it as a long i in the pronunciation. It is the earliest version I have seen and the only one other than the website.

Shy-rus is the only way. Anything else is uncivilized.

I hate civilization. Sheee'roosh it is.

George Shiras did much of his initial nightime flash photography work here in the Marquette area of Upper MI.
His name is pronounced Shy-rus.

My buddy Rick with half of his 2015 Colorado Shy-Russ moose

Are they shy or bold???

I've usually heard it pronounced "shy-rus".......... But heard a Cunuck pronounce it "sheer-us". I figured he was a Canuck, and didn't know better...........


Casey

I would've expected sheer-us-eh.

Ditto when I hung out in Wyoming. Next....Coues Deer. I heard it was supposed to sound like Cows deer and had something to do with Native Americans in the area where they are found. Fun stuff, we should start a "hooked on phonics" for hunters.

The deer was named for Harvard naturalist Elliot Coues, who did pronounce his name "cows". He wrote a book on ornithology and the first chapter was all about how to care for your shotgun. This was back in the days when collecting was the first step in systematic biology.

Systematic biologist sounds much better than old fat hunter....I'm gonna switch.

some people say it Shiras, I've always heard Shiras.
but the natives pronounce it Shiras.
I forwarded your question to a professor of linguistics at Auburn and his email follows.
"Shiras"
regards, Dr Talkywordy