How can I get an extension on the 14 days of Temporary Lodging Expense?

Each situation outside of the norm is reviewed. If you have specific circumstance preventing you from moving on or off base then you can request a waiver.

What throws me is that you say that your wife just PCS to your location, I assume you were in barracks, that too is yet an option if space is available, you would have to separate until available. You will be asked why you did not seek quarters prior to her arrival with her orders in hand. These factors may override any waiver as you appear to have had opportunity.

CPL (Join to see), you're limited to 14 days unless you're in one of the locations that's been approved for a standing extension (JBLM isn't on the list) or if there is a housing shortage (>98% occupancy for govt controlled/owned housing and <3% vacancy of rental property).

On thing to consider - often military servicemembers move between PDS locations in a shorter amount of time than they are authorized for travel. While you may be focused on the TLE, make sure you are counting all your authorized days to have per diem (keep in mind that TLE and per diem can't be paid at the same time though).
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* DoD 7000.14-R - Joint Financial Regulation (aka "the financial bible for the military") - https://comptroller.defense.gov/Portals/45/documents/fmr/current/07a/07a_68.pdf

If you have to get out of hotel try AF side of base, I see you are at a joint base. Back in the day when I was at Bragg, visiting quarters were much more expensive than Pope AFB, so I often stayed at Pope. Better service, too. Army had NAF civilians, so I guess they were more expensive. AF has airmen and airwomen that are in Force Support.