Too soon to request to relocate?

To keep this short the full time I got hired to a week ago blatantly lied to me about pay and so I can no longer move closer to the location. There is a location far closer to where I currently live and I was planning on requesting to transfer to that location stating three things.

• Apartment fell through due to unforseen circumstances (truth)
• I cannot sustainably continue to keep making the trips knowing that I won't be moving into the area for the foreseeable future (truth)
• Sudden decline of family members health would allow for me to care for them better (lie but they lied to me multiple times so who cares)

I plan on at least waiting roughly a month though so I can get my job certification. To show my "commitment" because I still have shadowing and a exam to do. Is this a fine idea?

(I plan on working here just long enough to find something better)