I use “compleat” to mean “quintessential”. That is, if I say someone is “the compleat (whatever)” I mean that he or she is the embodiment of the perfect example of the practitioner of his or her craft.
“Complete” means “it has all its parts”.
Am I using these correctly?
Yes. The distinction came about because of Izaak Walton’s famous book, The Compleat Angler, written back before spelling was standardized. So he chose to spell the word “Compleat”, and, even though the word’s spelling has since been standardized as “Complete”, the book has had such enduring impact that people continually borrow the title’s construction (say Maxim doing a story called, “The Compleat Bachelor”, etc.). So now “compleat” means “quintessential”, as you say.