Eeboo "Tree of Life" 1000pc - how do you protect a long-term WIP from kids and a cat?

I'm working on Eeboo's Tree of Life (1000 pieces) and quickly realizing my usual setup won't work.

I have two little helpers who love to "help" and a cat who's basically a furry wrecking ball. Normally I do 500piece puzzles on the dining table and finish them over a couple of evenings, but this one is taking longer because I only get short pockets of time after bedtime - and I keep getting distracted by Wordle and my phone.

So far I've done the edges and sorted pieces into color piles, but I keep having to clear the table for meals. I tried sliding the whole thing onto a big piece of cardboard, but it flexed and pieces popped loose. I also tried covering it with a sheet, which apparently was an invitation for the cat to nap right in the most important section.

What do other parents/pet owners do for a longterm WIP? Do you use puzzle boards or mats, stacking trays, foamcore bases, those rollup mats, or something else? How do you move the work safely without bending it and making pieces fall? Any good tricks for keeping pieces organized when you might not touch it for a day or two?

Bonus: do you let kids place pieces whenever they want, or do you set rules (only edges, only a tray, etc.)? I'd love tips, product recs, or even Ikeahacks.