Discard it as is. Roll it up and secure it with some string or tape.
Don't bother to put it into another piece of plastic (the bag) which adds to the plastic pollution of the planet.
Alternately, run a sharp knife down each of the rows of bubbles. Even cutting every-other row will decrease the volume significantly.
Is there a UPS store or FedEx outlet ...
I am thinking about using some kind of wire brush or metal curry comb.
There are some wire brushes in the shape of cylinder that you could simply bowl on that bubble wrap. If brush is sharp enough it should perforate the bubbles.
Please note that I've never tested that "lifehack" so it's just a guess.
I fought this same battle for years. In the end the cheapest and easiest solution was to change to a 'monochrome laser all-in-one'
I tossed out much more money in dried up ink over the years, then the cost of replacing the machine with a laser printer.
So now I only print black and white, and I am still using the toner that came with the machine 2 or 3 ...
The best remedy for clogging is to perform periodic maintenance by taking your vehicle out on the road to clean out the cobwebs every once in a while.
Note: I have a supply of shopping list paper (pre-printed) as a side benefit to this practice.
Give some thought to the test page. It should have some of each of the 4 colour inks, Cyan, Magenta, Yellow, and ...