How can I change my profile ID so I can make copies of my profile?
You can’t.
That’s disappointing and sad considering what all else can be modded like game-saves, save ID’s, gamerpics, XBL tenure and account info including XUID. 
After reading your original post again, it’s possible to just copy & paste your profile to use as a back-up if you make a mistake using the Achievement Unlocker for example but you can’t just change your IDs to replace/replicate anything on your profile.
XUID is changed from a modded console not a retail.
Why would you want to copy your profile anyway?
To have more than 1 account with same achievements and score. Also, I’m not sure if profile ID is the same when you join XBL. I read on a Modio thread about offline time-stamps being made null or all 0’s because the time on your console isn’t trusted to be accurate.
Here’s a link to the thread and yes, it is off-topic but I brought it up due to a recent post I made which only 1 person answered to.
http://www.game-tuts.com/community/f37/can-i-add-timestamps-offline-acquired-achievements-3929/