So how do you plan on screening them to know if they’re not married/attached?

Interesting you mention compassion, it's the compassion and empathy I feel for their wives that forms the mental block for me engaging with married men! To put it short, I believe Relationships should be about communication and honesty, that doesn't mean I don't feel for people who find themselves in dead bedroom situations, but sneaking behind someone's back and lying is not the answer to the problem that I want to support. I'd love to have the mindset of it being none of my business (and that's a valid viewpoint, I don't judge those who hold that viewpoint) but I can't justify that for myself when I'm directly supporting someone in their act of cheating.
Anyway, my question wasn't about the ethics of seeing married men, it's not a can of worms I wanted to open truthfully😂 I'm just aware that it's not uncommon for some to choose to exclude certain clients (by age, race, ability) and so I hoped that in a similar way, I could exclude married clients. Was just looking at the most practical and tactful way of achieving this 

It isn't our responsibility to police other people's relationship fidelity for them, nor is it our place to judge our clients personal circumstances. You are free to see or not see whoever you choose, but I would seriously think about whether escorting is a good fit for you, if you can't get your head around the idea of having sex with men who are in relationships/married.
If it's a moral line in the sand for you, then this job probably isn't for you and will eat away at you over time. The men you filter out with a "no married men" rule will be the honest ones who are probably decent clients, and the ones who have no issues lying to you and are a bit lower on the integrity scale are the ones you will end up with as a client base.

A large proportion of punters are married and visit escorts to avoid the messiness of affairs - and the ones who claim to be single are often the boundary pushers who are anything but single. If your priority is not being involved in someone else's cheating over cultivating a good client base for yourself, then I really would have a good think about this before you start taking bookings. This job isn't suitable for everyone and it may be that it isn't suitable for you.
We have no right to know our clients' relationship status, just as they have no right to know ours.

We as sex workers operate with alter egos to protect our identities so in effect we lie to our clients for the best reasons.  They can choose to do likewise and tell us they are single when not or divorced while married etc. Asking only unattached men to pay us would not be the wisest road to straightforward and stress free bookings. There's no way of knowing if they're being honest and I am visualising a booking with a single (according to him) man who may want to discuss your unusual mindset and it just seems to me it would spoil the whole meeting.

Some men are turned on at the thought of cheating and while that may be abhorrent to many it pays our bills.

Mmm, it's a question that plenty of married men are not going to answer honestly because they are going to think you might use it against them later via extortion.

What's the upside ever going to be to say 'yes'?

If you are determined to try to sniff them out, you might try saying that you sometimes enjoy threesomes with regular clients, 'meeting' them as a couple somewhere. There are going to be men who say my partner might well be up for that...

But really, as someone else has said, this is another reason why having a work persona who doesn't care is a good idea.

I just want to add that married clients can lie to their wives (of course I would never cheat darling) so they can just as easily lie to us. In a nutshell you would never know if they have a wife or partner.

This is not the job for you. It’s like trying to be a vegan butcher? I think maybe 10% of my clients are single. And they are usually not the most desirable clients also.

So do I.

If you were a paramedic, would you be fine to treat and potentially save the lives of drunk drivers, murderers and others who had done things or held views you find abhorrent, despite that being your job? Or if you were a prison officer, where your job would be to protect and ensure the safety, care and dignity of people who may have committed horrific crimes?

For argument's sake, being a sex worker providing services to a cheater is different from these examples, because in doing so the sex worker would be enabling the very behaviour that she finds abhorrent. The best fitting example I can think of would be a bar tender providing alcohol to someone who they know intends to drive themselves home afterwards.

Now obviously, I know by not providing services to men who cheat, I won't be stopping them cheating, because there'll just go and use another provider to do that. But at least I wouldn't be the one enabling that behaviour.