The thing that I don't understand about your experience as conveyed above is why they didn't accept your deedback. After all, you were willing to pay their almost unbelievably expensive $2500 to do so.
That makes little sense to me as to any Florida timeshare since, from my understanding via this site, as long as you don't object to their foreclosure (very important!), and they do foreclose, Westgate thereafter has zero ability to collect anything from you. There would be no deficiency judgment, no judgment for allegedly past due maintenance fees. All they'd get for presumably sky high legal bills would be the timeshare back.
Again, that's my understanding from what I've read on TUG as pertains to Florida timeshares. So by no means is that legal advice and I advise anyone similarly situated to consult a Florida attorney with knowledge about Florida timeshare foreclosure law.
But perhaps you've stated an objection to a foreclosure in some way. "I don't owe this or that because you defrauded me" (an objection) or some such message may have been communicated to them. In that case, they can conceivably get a judgment for unpaid maintenance fees plus contractual fees and penalties plus attorney's fees. And that unquestionably WOULD have a significant impact upon your credit record. Not to mention your being pursued for the collection of that judgment.
To me, if a resort knows the law, and knows that you know the law (perhaps via an attorney's intervention), and as a result knows that they would have no recourse against a non-objecting non-payer, it seems they would be eager to take back the deed at no charge rather than wasting their legal fees.
If you might possibly maybe have objected, and that's why they're treating you so confidently and boldly, again I suggest you speak to an appropriate attorney. Perhaps (and this is mere speculation) he can change the messaging to:
"My client acknowledges that your computation of past due maintenance fees plus other fees is 100% accurate. However, he/she is unable to pay those fees. He/she would have no objection to your foreclosing upon him/her. But perhaps you might prefer to do a deedback which he/she's willing to do without hesitation".