Canonically ? which version of the canon are you referring to ? One of the 4-5 Different batmans from the movies or one of the ones from the several animated movies ? or from one of the several TV Series Including 2D and 3D animations
. If a billionaire was paying women of low morals to tell others he had sex with them then the rumors of him doing that would be more widespread, the Tabloids would have fun guessing what's he trying to hide.
Sure most of the stuff with Batman in it is intended for younger audiences so his sexual encounters are implied rather than graphically shown like in Game of Thrones. Depending on the storyline and the version (comics/TV show/movie/etc) he has been oficially in love with a lot of Heroines and Villainesses. Would Love to see a proper R-rated movie with Batman in it like the Logan Movie, Ben Affleck's done a pretty decent job at playing at-least the batman part of the role, those fight scenes in BVS were delicious ... not the killing part though. Just Batman Crippling random thugs left and right.
Just because SR7 checks here sometimes and it may help add flavour text (Lex knowing who Batman and Superman are and spying on shit tons of stuff)
Here are some of the various people Bruce Wayne in various cannons has slept with that are relevant in the context of who is n the game and who could be in the game and it make some sense
Black Canary (yeh Green Arrow got Batman's sloppy second basically, though this was comics but also apparently mentioned in passing in the Justice League cartoon)
Zatanna
Lois Lane (yes she's not only dated Lex Luthor before but Bruce Wayne too, in the comics they were getting married, in the animated stuff well they were dating and sleeping in the same bed in The Batman Superman Movie.)
Wonder Woman (apparently in the comics it's implied they did more than date, though in the animated series they have about 2 dates but still have feelings for one another)
Andrea Beaumont - from the animated film
Batman: Mark of the Phantasm
Talia Al Ghul - in the new DC animated and the comics he had a son with her Damien Wayne.
Selina Kyle - In Batman Retruns and the comics.
Batgirl (Barbara Gordon) - DC animated's version of The Killing Joke & mentioned in Batman Beyond.
Poison Ivy - Comics only
BEKKA - in the animated series I believe and comics though versions vary her power is to make people fall in love with her but she also falls in love with them.
The Joker - (Flashpoint continuity I'm not saying anything else other than Batman isn't Bruce Wyane and the Flashpoint Joker is a woman)