You should consider the possibility of using the packaging materials to sequester carbon dioxide from the atmosphere.
Cardboard is made from trees that have taken carbon dioxide away from the atmosphere. True, the cardboard manufacturing process releases some of this carbon dioxide back to the atmosphere, but much of it stays in the cardboard.
Plastic is made from oil. Oil comes underground from fossil fuel reserves. If it wasn't used to make plastic, it would be burnt in internal combustion engines.
By using oil and cardboard and burying them in a responsible landfill that doesn't contribute to the microplastics problem, you will:
- ensure the carbon in the plastic / oil will be returned underground from whence it came
- ensure some of the carbon dioxide trees sequestered will be buried underground instead of being released back to the carbon cycle
Now, I don't know if it's feasible everywhere to ensure your waste will be buried in a responsible landfill. Some landfills contribute to the microplastics problem; other locations burn the waste back to energy.
Even if you live in an area where waste is incinerated, you get at least dual use out of it: first, it is the wine packaging, then it will be burned to energy that most likely will create electricity. Much better than burning the oil directly to energy in engines, because that gives only single use of the source materials.
If you have the option to decide where your waste will end up being (for small amounts of waste you don't have the option), and are unwilling to consider landfills, consider a location where district heating is widely used. The efficiency of cogenerating electricity and heat is over 90%; the efficiency of generating just electricity is around 40%. Locations that need heating energy in winter can thus use more of the energy in the waste for useful purposes. So, you get triple use: first wine packaging, then heat and electricity. (I happen to live in an area with district heating, and much of it is cogenerated along with electricity from waste.)
Apart from that, if the wine bottles are reused, they will need to be cleaned in an industrial process before returning them to the second use to function as wine bottles again. This cleaning process may be environmentally harmful due to requiring chemicals. I certainly cannot prove it's more harmful than the plastic inside the cardboard packaging, but just something to consider.
Now, you won't solve climate change by drinking wine, of course! The packaging stores so little carbon (it's very lightweight) that is will have negligible impact. I would assume this light weight of the packaging suggests that it might be more wise to focus about the environmental impact of the contents and not of the packaging.
Also, you correctly noticed boxed wine stays fresh longer. Even if only some of it is left over, it probably matters more than the packaging.