The very first thing we did when we got our MHC was to remove that mesh bag and used that space for storage. As a laundry hamper, it never worked well for us. As a storage space for extra toilet paper, paper towels, a couple fans and a couple ceramic space heaters, it's perfect.
What do we do with our laundry waiting? It goes directly into the washing machine as we have a washer and dryer on-board. No need for a laundry hamper any more. Once the machine is full, simply add soap and turn it out. We always wash in cold water and lights and darks are washed together. Only for specific items of clothing are they washed separate so colors won't bleed, or grey. Occasionally, we'll pull everything out and run 2 loads, especially if we want to bleach out whites.
We did not always have the on-board washer and dryer. Our previous Outback travel trailer had a laundry hamper also, and it was a pain in the neck. We ended up using baskets in the closet for dirty laundry as the baskets were easier to grab and go to go to a laundry mat.
Each person does it different, for sure. If you can't find a mesh and really insist on using that space as a laundry hamper, you can always pick up some cloth from a fabric store and fashion a bag of yourself. Can't sew? Then use a staple gun and run lots of staples to hold the cloth together to make a bag? There are ways to do it if you can't find your most desired choice.
Two campers ago, we used my old Army "laundry bags". I had 3 of them, drab military green, but they hung on my bunk for several years and served well. They worked out great, but they were never used inside the actual laundry hamper space. Once those bags had stuff inside of them, they were near impossible to pull out. The trim on the edges blocked a stuffed bag from being pulled out. It was easier to just keep them in a closet.
There are alternative ways. Be creative!