Lycra is the Du Pont trade name for Spandex.
Spandex is made from a fibre incorporating elastane, the elastane makes it stretchy.
You can get many types of Spandex, including Nylon Spandex, Polyester Spandex and cotton Spandex.
As you are allergic to Polyester you will also be allergic to Polyester Spandex.
You have said that many swimsuits have Polyester linings, this would be the white, cream or even black absorbent linings used by Speedo etc. Some brands such as Aussibum line their briefs in the same fabric as the suit itself. The Aussibum Classic is made from Nylon rather than Spandex and is lined in the same fabric. Perhaps you would like this:
http://www.aussiebum.com/swimwear/cl....5-citrus/2949
I have the citrus ones, they are just opaque enough to wear in public, the lighter colours are transparent when wet so beware.