Though he only time travels when bathing maybe he has a different strain (as CDD in The Time Traveler's Wife
is only just being begun to be studied it is entirely possible that there is more than one strain) where the time travel is brought on by not just general stress but work related stress in particular - and has Lucius is bath engineer there is your connection. Also this is why he keeps reappearing in bath related places when he does time travel as it is all relative in his case rather than Henry's more sporadic strain (though even Henry's tended to focus a lot on certain life events - mostly his mothers death - so Lucius' case is just more precise). Also the limitation of losing all your clothing when you time travel isn't an issue for Lucius because he's always naked when it happens anyway because he's in a bath! Now this does mean Lucius is DAMN lucky compared to Henry and even Alma from TTTW
who never had this exact a set of scenarios, if they had they probably would have had easier lives.
- Jossed for the movie version: there is a conductor controlling his time travel. For the manga, it is going to be more reasonable.