I have not put loaded panniers inside of a mesh bag, but I have used mesh duffels for empty panniers and their contents. Amtrak, the local agent was an extreme stickler for the rules, would not allow two panniers to be strapped together to count as one carry on, etc. So, for my Pacific Coast trip where we started and ended with Amtrak, I used a large mesh duffel and a medium sized mesh duffle, the duffels when empty were in the bottom of a pannier out of the way. I my case, the reason for mesh was simply it had the lightest weight. I emptied out the panniers to fit in the duffles. The empty panniers were also in the duffles.
Photo above at the start of the trip, the two duffles closest to the camera are mine, the bottom one (blue ends) was checked and the top one (red ends) was a carryon bag. The other bags belonged to my touring partner.
My carry on bags for the trip home, photo below. The same medium sized duffle as in the above photo, plus a shopping bag. I think the shopping bag was a reusable one from Safeway?
Do not rule out Ikea bags, they make some that are cheap and could be robust enough to use, I am thinking of the blue ones.
I have never used this particular bag, but I bought one just in case it comes in handy some day. It was cheap enough to just have it on the shelf in case I need it some day.
https://www.ikea.com/ca/en/p/frakta-...blue-90149148/
ADDENDUM, November 27, 2023:
The Ikea bag I cited in the paragraph above is no longer available. I tried to buy one a week ago, sales staff said they were no longer available.