This is derived from the phrase "to weather a/the storm"
To weather a storm is to get though it as best you can, and make it through to the other side.
This metaphor can then get extended into "to weather the financial storm".
At which point, it seems, it has been mutated into "financially weathering".
The second "weatherization" in the example sentence is to do with weatherisation of homes. That is, cutting down on the energy usage of a dwelling using various energy conservation measures by fitting: insulation; draught-exclusion; double-glazing; energy-efficient lighting; etc.
In addition to energy-conservation measures, it seems the use is being extended to domestic energy generation measures, and other things which are not weather-related.
To the sentence means:
Surviving the cost-pressures of upgrading your home's energy-conservation measures.