As a professional carpet cleaner in Auckland I can add to this conversation from a different angle...
The thing you need to know is that not all carpet cleaners are created equal. Every city has 3 types of carpet cleaner available.
Mr Cheap: Your $160 quote is from Mr Cheap. Mr cheap is a churn and burn operation. He quotes low prices to win work, then has to rush through 4, 5, or even 6, jobs in a day to make any profit. He mostly just goes "through the motions" of cleaning. He doesn't go slowly to ensure proper steam injection. he doesn't go back over the wet carpet 3 times to extract left over water to make the carpet dry faster. He doesn't treat different carpets in different ways depending on type and other factors. He might not do stain removal, he might not use the best or any chemicals. He certainly doesn't get lots of repeat business or good online reviews. Lastly, there's a high chance he'll be using a portable machine that comes into the house. These machines are WEAK compared to high powered truck mounted cleaning machines.
Mr Middle: Mr middle can be a good option for many home owners and even more so for tenants. He is charging a around 50% more than mr Cheap. Because of this, he can make the same money in 3-4 jobs as the other guy makes in 6 jobs. Because of this, he takes a bit more time and care to do the job properly. He balances service with price and gets most little marks and stains as he goes. He should do the job properly without going to extremes in regards to customer service or fancy extras.
Mr Premium: Your landlord being the owner of the property and the one who cares most is going to call Mr Premium. Mr Premium is amazing if you can afford him. His vans are well branded and in good condition. He lays on the extra special treatment to justify the price. He'll do all sorts of stuff like, pre-vacuum before cleaning, do extra treatments on high traffic areas, he will use commercial grade deodorizers to eliminate smells and other mess from pets, kids, and spills. He has a wide range of (sometimes) expensive chemicals to treat and get rid of every kind of stain. He is fully insured, properly trained, and WON'T damage the property by dragging hoses against walls.
So, as far as the price difference goes, I hope the above is enlightening...
Now, for your specific situation.
The Dog: as sweet as he is and well trained and not shedding etc... it simply doesn't matter. if the dog has been hanging out on the carpet he has put into it all sorts of doggy oils, doggy sweat and doggy slobber. The second the steam from the cleaner hits the carpet all you're going to smell is WET DOG. The use of special chemicals to break down these organic compounds from the dog is a must if the property is changing hands. You could really ruin the day of someone with dog allergies if this is not done properly.
What if you hire Mr Cheap and he damages the carpet?
What if you hire Mr Cheap, and he does such and average job the landlord is forced to bring in his on guy... now you've paid to clean the carpet twice! (i see this often)
I hope you sort out your situation, armed with this new information perhaps it may be easier to understand the difference in price and why your landlord goes for premium while you go for Mr Cheap...
I think the key for anyone else in this situation is Mr Middle. Check the online reviews and ensure they're reputable and get a good job done without the massive price tag.