If you are growing beans and courgettes raise them in pots and then plant them out. This will mean that you do not need a fine cat attractive seedbed. Cats prefer dry soil. If you can keep the soil damp they do not usually use it
Actually, I have been growing mine in pots--I'm thinking of planting them out in the next week or so (hence my concern about the poop now). That's really interesting about the damp soil--I'll have to test that one this week, because if it works on my cats, then I won't need to worry about anything else.
I plan to let my runners go up an arch, so I don't know how effective mesh would be on something like that....but I'm definitely bearing it in mind if all else fails.
Thanks for the advice, everyone.