In the mystery genre the readers are specific about their likes. Cozy readers want a safe environment with familiar icons in it.
A sidekick who's a dom is pushing it. Ditto for your detective going to her. It also reinforces the stereotype that all Brits are sexual kinks. Maybe they
are, but benefit of a doubt and all that. (And no I'm not saying this is a kinky thing in these politically correct days, but much of your potential readership might.)
You can keep it humorous--points to you if you manage that!--but if you intend an open-ended series, that's going to be hard to maintain as the characters grow and develop.
It will also be a hard sell. An editor might take a chance on it, but they know their readers, and most will not.
Perhaps your first choice for "quirky" should be to find something other than a sexual quirk to fill out your menu.
Speaking as a fuddy-duddy in training, I question why he would put up with what sounds to be a horribly unpleasant person. Getting sexual gratification from a woman is as good a reason as any for some men to put up with put-downs and sarcasm, but take sex out of the picture and you've got an unbelievable situation.
If your she was a he, and your detective straight, would he tolerate such obnoxious behavior from a male helper?
Swap genders. If your detective was female and the uber-sarcastic one delivering put-downs male, you'd have feminist groups picketing your office if not burning your books.
Perhaps some more beta readers are needed to pinpoint what the problem is.
A good mystery detective should be someone a reader can admire and aspire to be like. We would all love to be Sherlock Holmes, but Holmes was not so nice, being a rude, messy drug addict with few social skills. Watson was a likable fellow who made Holmes likable to others.
A mildly sarcastic pipe-smoking professor is one thing, but having him going to a dominatrix will squick most cozy readers out. There are plenty of fuddy-duddy readers who will lose respect for him, and you can't have that for a hero.
If you want to keep the characters as-is, I would suggest he not be the girl's customer. To him it can be just a job that he's aware of and separate from whatever partnership they form to solve crimes.
I can give you a safe quirk: he can be addicted to Scooby-Doo cartoons. I had a roomie for a time who was a bi-sexual escort when he wasn't bartending, and was as obnoxious as they come--just not to me, he was smart enough not to do that--and he could not get enough of Scooby and the gang.
Your professor can draw parallels between his cases and Scooby mysteries the way Remington Steele did between his cases and his beloved movies. (There's a good quirk.)
Quirky on TV: Monk, House (He's Sherlock Holmes with a drug addiction, a medical degree and Watson's limp), Columbo (just one more thing...), there are more, but I have to write today.
You might want to read this book by
Delores French as it has a chapter on her turn as a dom. It's pretty funny, but you get the vibe that she had a touch of contempt for the clients.
A sidekick as sarcastic as you say would also very likely have contempt for the professor, not be able to resist putting him down for it--their "sessions" bleeding into daily life--and she'd have a lot of blackmail material!
An alternative job for her could be as a stripper, but research that before taking the leap. It's not the cleaned up stuff we see on cop shows; they have to deal with sleazy drunks, addicts, bouncers, late hours, and a short shelf life going from club to club.
One of my writer buds sold a cozy that had murder at a strip club. She interviewed several real-life strippers and noticed a common thread of hating men in many of them. They were often alcoholics, chain smokers and/or drug addicts. They'd been victims of abuse and battering from the men in their pasts. One had a strong wish to beat her father to death. Not too cozy!
Those cozy detectives with their quirks, including cats, gardening, cooking, knitting, are mainstay reading for grannies wanting a
safe morality play. If they want sex in the mix they'll pick up a hard-boiled or a thriller or cross the aisle for a romance!
You might want to read these two books by Dilys Winn.
Murder Ink; Murderess Ink.
They should be in the library, or you can interlibrary loan them. They taught me
much about the genre.
Good luck!