Problem with negotiating notice

Hi folks, I have been searching for a job over a year in the field of Data Engineering on GCP.Whenever I receive call from recruiters I was being completely honest with my notice period(NP) that it was 60 days but recruiters were looking for immediate joiners from 15 to 30 days. After informing the recruiters that my NP is 60 days I wouldn’t receive the next call or a schedule for an interview. This was the case for over 6-8 months. Realising that I’m running out of time, I decided to lie on my NP that it was 30 days or I’m an immediate joiner and I could also negotiate the notice with my current company. I had attended few interviews after bluffing on my notice and finally cleared it 3 weeks ago. The new employer is asking me to join in the first week of Sept but my notice ends on first week of Oct. I tried negotiating the notice with my present employer by talking to my manager and HR but they are adamant and won’t relieve me in the first week of September. They are also not accepting the buyout option which is available on the Offer Letter. If I tell the truth now to new employer then they will lose trust and integrity in me. I asked for an extension of few days but the new company is not ready since they want to kick off the project immediately. How to navigate this situation ? How to convince either my old or new employer ?