Can this be done?

No. you cannot dye anything that cannot be washed. It can't be done professionally, either. "Dry clean only" also means "you can't dye this."

Some garments that are marked "dry clean only" can actually be washed, but never if it has a separate outside and lining. The lining and the outside layer will shrink at different rates, ruining the shape of the dress.

Furthermore, polyester can only be dyed with a special type of dye called Disperse Dye, which requires that you boil the dress in it for an hour, but even a minute of boiling will ruin the lycra, which is a thermoplastic polyurethane. See http://www.pburch.net/dyeing/dispersedye.shtml.

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