First step, let them know that the nuisance barking is not on.
That is, barking for long periods without break, barking outside certain hours (11pm to 7am probably or thereabouts)...
I would not mention their business, and I would not expect a resolution but.. the council will ask you if you've talked to them, this shows you have.
Then you can move forward with an actual noise complaint - if they are not meant to be running their business from home and it is contributing to the noise issue directly (ie its dogs coming in for grooming that are barking, rather than their own dogs) then theres a chance that will be stopped.
There is a chance they are permitted to run the business from home, many people do, as it shouldn't be noisey and it shouldn't increase traffic, or produce waste water/products outside the usual household stuff.. and those are generally the main concerns about home businesses.
If it is legit.. and it is clients dogs barking and that barking is not constant just.. annoying or its not clients dogs its just their dogs having the odd woof, like yours dogs only more frequently, and theres more of them.. there is likely not a lot you can do there, they are allowed to WFH just as you are.. homes are not really designed to be workplaces... they're not quiet.
I do suspect they can be warned however about the overnight barking, at the very least.