Funny you asked and since I allowed myself to get hooked on surf fishing, this very thought has been on my mind too.
So, here's my thinking. A 30ish qt. el cheapo cooler with a hard plastic liner. However, the hard plastic liner may be difficult to find so I'm thinking el cheapo plastic tote, with lid, that fits inside the cooler.
The tote is for any stupid fish I catch. 🤣 So in my case, the tote should be exceptionally..... small.
OK enough of the jokes and I think you see what I'm sayin.
For the ice melting and leaking water, first line the cooler with a plastic trash bag and pack the ice under and around the plastic tote or just fill the tote with ice, which is what I could do because the ice would be the only thing in my tote.😝
The lid for the tote could serve as a "cutting" board for bait.
My luck the only "fishy smell" would stem from cutting live bait on the tote lid but still, a wash of chlorinated water (Clorox) should help with el stinko.
Just thinking out loud but I hope it gives you some food for thought.