Unclear on issue. will say right now that very few sofa beds give you both a comfortable sofa, AND a comfortable bed. Unless high quality, they are the worst of "both worlds". You won' be able to fold the bed, or will have to store any solid thing you put over the 'bars". Pad the bars with some very high density foam cut to size and hope for the best. Which nothing will be.

Use bunkie boards. Store in closet.

Before I bought a new sofa bed I used to put plywood between the bars and the mattress. it was a typical fold out hide a bed (which I will never buy again and would never recommend) It was surprisingly helpful though, but a real pain as you can imagine. If it's in your budget have a look at The Sofa Bed Store, I'm happy with what I got there - no bars to deal with and a solid thick mattress, no complaints from guests, my sister even purchased one after me. But if you are in a pinch and a new one isn't in the current budget, give the plywood a try (if you happen to have a sheet sitting around your garage).