I am this person with certain words. I try to keep the pronunciation as close to the native language's as possible if I am familiar with it. I try to say "root" for "route" and "foy-ey" for "foyer". Pronouncing "croissant" with the French pronunciation is awkward though, so I usually say it sarcastically. The "e" is "Porsche" always pronounced; even by English pronunciation rules, a vowel preceded by
three consosants is pronounced and not silent! Likewise with "Schadenfreude".
I care more about pronunciation and clarity than accent (they are related, but distinct).
I notice things about my own native accent that I don't like, but it sounds a bit weird when I try to change things. For example, I find it ambiguous that I and most Americans use flaps for t's. Latter and ladder, hearty and hardy, leader and liter, let on and led on, metal and medal. By using hard t's in those cases, I end up sounding British or Irish for that one word unintentionally. I also recently learned that New Yorkers distinguish between "cot" and "caught" and between "merry", "marry", and "Mary", which is pretty cool, but I don't think I'm ready to adopt a New York accent.
Humans are just plain weird.
Not sure if this fits the thread, but I have an acquaintance who speaks in a general American accent but will use a British accent on a seemingly-random word. Like, when saying "can't" he'll pronounce the "a" as the "o" in "hot". He's in his early-mid twenties, so maybe he's still acclimating to his vocal chords. 
this has nothing to do with British English
What would you say that it is, then?
I've never seen or heard that person, how the **** would I know?
The "o" in "hot" may not be exactly the right phoneme (or it's different in your accent), but Brits tend to change the vowel sound between "can" and "can't" and use a hard "t" sound; whereas, American pronounce "can" and "can't" almost exactly the same (with maybe a very short glottal stop at the end for the "t") but just put stress on the negative form (which makes it very hard to tell the difference between the positive and negative sometimes even as a native AE speaker). I personally just try to add a hard "t" at the end to make it clear I'm using the negative form.
Good video on the subject (link with timestamp to relevant part 13:34):