Even though I've now gracefully retired as a soul DJ and collector, on and off for 27 years, I can speak from past experience.
If I had a record in my box which was requested, I'd do my best to fit it into my set, providing the request wasn't unreasonable, or was otherwise likely to clear the dance floor (ie, American Pie by Don McLean!). If I didn't have what they wanted, I'd be upfront and say "sorry - I don't have it".
But then you do get some rude bastards around, who tell you what they want to hear, and generally have the manners of a pig. Those requests were always ignored.
In the early '80s, I was hassled by a leading thug of a then notorious scooter club to play a request, and to dedicate it to them (thankfully it wasn't Deutschland Uber Alles). I ignored the request twice, as they were always getting clubs closed down, and on his third attempt, preceded his request with "Look - unless you want the place smashed up...............".
Catch 22 situation or what? Just to shut them up, I ensured the next record was their request. They left me alone for the rest of my set and, soon afterwards, they got the club closed down that same night, when one of their "soldiers" glassed some girl in the club.