How to combine multiple bags into a single checked item?

You can read in https://travel.stackexchange.com/a/42030/4188

I was offered a giant, durable, resealable (great for security checks), clear plastic bag, similar to a trash bag but thicker, with the airline's logo on it, to prevent any straps or hooks from getting caught in the baggage handling. Since that flight I've found that most legacy airlines have these.

You can call ahead to make sure they have one at both airports (I presume you will be checking bags at two airports, one the start, one the end) but the same answer continues with:

I've often encountered check-in agents that didn't know their airline had these, but in every case they asked their colleagues or manager and discovered they did. I've never been charged for one. I've gotten these in Europe, Asia and North America.

Or you can buy for yourself. These are called contractor bags (or rubble sacks or builders sack) and their thickness in the USA is measured in mils which is a thousandth of an inch. Elsewhere it is measured in microns which is a thousandth of a millimetre. Thus a 100 micron bag is a 4 mil bag which is thick enough. I used 3 mil bags for protecting my plane side checked backpack and it was fine (but note plane side is not the same ordeal as a checked in bag by far).

I also found a A Buying Guide for Rubble Sacks which looks quite useful.

I have always just used a simple nylon duffel bag. The panniers fit inside for the flights and the duffel gets folded and strapped to the bike rack while pedaling. The duffel occasionally get used for storing items as I ride that need some protection from the elements.

The duffel has a big advantage over plastic bags as it has handles for easier carrying (by you and baggage handlers), handles which also provide an attachment point for luggage tags and ID tags. It also tends to be stronger, less rip prone than a plastic bag. Plus you always have it with you, so no hunting around for new ones every time you fly.