It would be better if you could wait just a bit until the weather gets a little cooler. However, crApe myrtles are pretty tough and are pretty likely to survive even if transplanted now. The bigger the rootball you can get, and the better you carefully maintain proper moisture levels after transplant, the better chance there is of success.

Here is a link that might be useful: Planting a Tree or Shrub

Totally agree.

But... are you sure the plants are going to be destroyed and not just the buildings? If whoever owns the land plans to rebuild within the existing landscape, that could be a problem. But it sounds like maybe eminent domain has relocated some residences to widen a road? If so, I would try to save them too, (assuming I could convince someone else to do the actual digging, I can't anymore.) No reason to not try.

Temporary mid-day shade can also be extremely helpful when replanting "out of season."

you gotta do.. what you gotta do.. when you can do it ...

find large pots [many nurseries have recycle piles].. dig them up ... pot them .. and put them in FULL SHADE.. until the proper planting season ...

bloom will probably immediately fail.. but that is not important..

water deeply.. and let near dry in between ...

and as noted.. get permission.. do not presume ...

the potting media might be important.. if you have a budget for such .... soil in pots can be tricky. but if this is short term.. it will probably work ...

i would then plant them.. when day heat falls.. and night temps are cool.. but will defer to anyone in TX as to timing ...

ken