What did Lovejoy think happened between Rose and Jack on the deck?

I watched Titanic for the first time recently and got to the scene shortly after Jack saved Rose from falling off the railing of the ship where Lovejoy notices a hole in Rose's story:

You'll want to tie those.

(Jack looks at his shoes)

Interesting that the young lady slipped so mighty all of a sudden and you still had time to take off your jacket and shoes. Mmmm?

What confuses me here is what Lovejoy thinks he knows about Rose and Jack at this point.

My first time through the movie, I thought Lovejoy was hinting that he saw through Rose's story that Jack saved Rose from slipping and was going with the guard's assumption that Jack was forcing himself on Rose. Which means, through the rest of the movie, Lovejoy thinks Jack attempted to rape Rose and Jack probably knows this as well since Lovejoy told Jack about his suspicions. If anything, it makes Lovejoy the only person in the group interested in Rose's safety since he was the only one to try to look past Rose's obvious lie, which Rose's mother and betrothed couldn't be bothered to do.

But that reading seems at odds with the movie's treatment of Lovejoy as a villain and a toadie to Cal, since he gets to be the one to leave Jack to die painfully after the iceberg hit. He'd have motive to do that if he thought Jack tried to rape Rose, but since neither party talks about the incident I can't tell if that's even the reason for the attempted murder.

Is there any evidence what Lovejoy inferred from Jack's clothing?

He knows there is more to the story

I think you are right that he sees through their story, but I think you are wrong in assuming that he concludes that it must mean the rape story was true. After all why would Rose defend her rapist when he was already in handcuffs and had multiple "witnesses"? It also doesn't play with the fact that Rose and Jack spend quite a bit of time together for the rest of the voyage, which wouldn't make sense if Jack was trying to rape her.

We are meant to conclude that he knows there is more to the story than they are letting on. At most, he may already suspect that they are carrying on an inappropriate relationship.

Later he is simply protecting his employer's interest

As I mentioned, Jack and Rose start spending a lot of time together during the voyage which does not go unnoticed by any of Rose's party. Lovejoy's job at this point is simply to protect the interests of his employer, Cal. He does this by shutting down any and all attempts by Jack to contact or go near Rose.