Chabad believes that anyone with a Jewish soul wants to connect to G-d by doing mitzvot.
Judaism also believes that non-Jews are able to connect to G-d by doing the 7 Noahide laws, so they don't need to become Jewish. (non- Jews get to connect with G-d AND eat bacon! Lucky them)
I don't know any Chabad people who believe converts are a lower status. Every Shavuot we read the story of Ruth, a convert, who was the great grandmother of King David, to remind us that ALL jews are converts (we took on Judaism at Mount Sinai) and that great people come from converts.
The only time I have seen negative attitudes towards converts it is from non religious Jews who are less educated.
One of my parents is a convert in fact, and I never got any negativity from people about it. I think one of my parents in law thought is was cool when our shidduch was suggested.
Once in school a girl came up to me and asked 'Is it true that your parent is a convert?' and I said yes. Only time I remember it being raised.
I once got a really stupid reaction that came back to me - it was not said to me. Somebody heard my parent is a convert and said, Oh, that is why Jewishbookwork is so pretty! (this was back when I was a lot slimmer!!!)
Really makes me angry that attitude, that Jewish women are not pretty. (as seen in hollywood movies where the Jewish heroine is NEVER played by a Jewish actor) I think I look pretty Jewish anyway, athough I do resemble my convert parent.