Guess Go Go Van might be a good option.  They can enter DB and drop you and your pet at your door, and the van have more space than Taxi.  they shall cost a little more tho.  You can get a quote from them and call to inquire.

Dont think Airport bus allows pet?

I guess it's too late to reply since the dog must have been picked up already. It would have been nice to know what option you used.

I didn't think a normal taxi would allow dogs in their car. Problem with dogs is hygiene, drool, fur, smell. It's very inconvenient for the driver and I think animals are not allowed on public transportation (bus, MTR, taxi)

They may make exceptions for toy-sized dogs which can fit in a bag and be of minimal exposure.

Airport bus should work but dog would need to be in its crate and stored with other luggage under the bus compartment. Doesn't sound nice.

Other options:

GoGoVan
https://www.gogovan.com.hk

LaLa Move
https://www.lalamove.com/hongkong/en/home

Uber
https://www.uber.com/en-HK/cities/hong-kong/

SynCab (for dogs, use MPT: Multi-Purpose Taxi ($5 per animal or light luggage)
http://syncab.com.hk/en/