It does no harm to tell MoM about the intention to apply for PR, the interview date (you can check regularly if earlier slots become available due to cancellations, and then change your appointment!) and that you'd need to bridge the gap until becoming PR.
The PEP is not renewable.
On the face of it your PR application would seem to have good chances for approval - but if you apply without your sons (who are still minors under Singapore law, so this will be seen as an attempt to avoid NS obligations), it will weigh heavily against you. If you're approved without them, their current Singapore passes will not be renewed any more, and the whole issue might well lead to PR rejection for the whole family - NS is a VERY sensitive issue in Singapore and avoiding it is not a good idea if you still need any official goodwill.