I believe you can take a Specialist exam whenever you want to regardless of Certification state.
You will get the Accreditation.
But your CCSE will not be extended by one year because it has expired.
Your CCTE Accreditation would also not contribute to achieving the CCSM Certification because it would not be passed while you hade a valid CCSE Certification.
In other words:
If your CCSE was still valid then the CCTE exam (when passed) would add 1 year to your CCSE certification period and then a second Specialist exam passed would award you the CCSM Certification.
If you take and pass the CCSE exam again before you take the CCTE exam then your CCSE will be valid again for 2 years and when you pass the CCTE exam your CCSE will be extended to be valid for a total of 3 years.
You would then have a valid CCSE Certification and the CCTE Accreditation.
You would also be one Specialist exam away from CCSM.
The answers are all in here:
https://support.checkpoint.com/results/sk/sk163417
Core = CCSA and CCSE
"Certification Extensions & Renewals
- Pass an Infinity Specialist Accreditation (ISA) to extend a valid Core certification for 1-year. (Does not apply to the CCSA certification)
- The ISA must be achieved after a Core certification to qualify for the extension.
- A CCSE may be extended one time. A second ISA will award the Accreditation and the CCSM Certification! (The CCSE must be current, not expired)"
"Infinity Specialist Accreditations
Check Point’s Infinity Specialist Accreditations are designed to enable Security professionals to demonstrate that they understand the concepts and skills needed to administer, deploy and manage specific Security solutions within the Check Point Infinity Cyber Security Architecture.
Infinity Specialist Accreditations are also the building blocks to the Security Master and Security Master Elite certifications. The selected Hacking Point exams shown below can also count as an ISA towards certification extensions."
Certification Path to Cyber Security Mastership
Check Point Certified Security Master
With Check Point’s path to Security Mastery, you need to achieve 2 Infinity Specialist Accreditations after the CCSE.
