How to convince suspension cells to adhere more tightly?
My main concern here is why you are using suspension-culture selected cells for an adherent cell-based assay? Is it possible to obtain a clone of adherent HT1080? Depending on what you are assaying with your protein of interest, the cell biology may be altered changing from suspension/adherent culture systems.

Here are my experiences using PC12 cells in culture, which are weakly adherent cells, with a preference to grow in clumps, rather than monolayers. I have found that these cells have a preference for specific tissue plastic; (without mentioning names), these cells will adhere on some poly-D-lysine coated plates while they won't adhere on others. I routinely perform immunofluorescence microscopy with these cells, and grow them on glass cover slips which I have skim-coated with rat tail type I collagen. This coating provides minor background in the green range and only when imaging close to the glass. With a bright fluorophore (e.g., AlexaFluor or cyanin dyes for fixed cell; enhanced GFP for live cells), this backgorund is not a problem for quantitative microscopy.

Lastly, for secretion assays we use high-glucose DMEM supplemented with 1% dialyzed FBS. If memory serves, this retains all serum components less than ~15 kDa, which includes various growth factors. I have observed these cells continue to live for up to one day (and maybe longer?) in supplemented medium, while they die after a few hours without. This may also help with non-specific binding you may be worried about with drug addition.