I just did that same mod on my new Forest River trailer with 1-3/8 door. Yes, they do make "mobile home" sets and the only difference is the doohicky that slides into the door with the plunger doohicky (sorry, don't know what that thing is called) is not screwed in like you see in homes. A 70 year old old timer at ACE Hardware filled me in on that. The mobile home set has a round ring and it simply presses in. Yours is probably this way, meaning it just presses in, but don't worry.
Do this: take your handle set down to real hardware store like ACE and look for an old timer. I'll bet you can leave the plunger doohicky in the door and not remove it at all and just install a new locking handle set.
That's what I did when the old timer at ACE explained the difference and told me not to spend the extra 3 bucks on the "motorhome" set but by the regular home set and leave the other part in the door and just slide my new locking handle in and re-used it. After about a 2 minute install job, I discovered he was right.
Maybe this will work for you, I don't know, but that's what I did and all's well. As to the width of the door, don't worry about it as the new handle will have enough adjustment in and out to accommodate just about anything. Let us know how it turns out.
By the way, the handle I bought was made by Kwikset.