A couple options to consider. . .
If the ottoman is to be easily moved about for the most comfortable "feet-up" use, a lot of weight is going to e a bother. change to drawer glides that are shorter and do not allow the drawer to be opened to the tipping point; a false back in the drawer would avoid stuff moving to the back and being difficult to access.
If the drawer's capacity outweighs (sorry I couldn't resist) the need for mobility, run to the scrap iron yard and have them cut you some 3/8" plate to a size that would fit inside the carcass, top to bottom and about halfway from front to back. Drill some holes and mount these inside the carcass between the drawer sides and the interior.