To apply for FLR you would need to do so under a specific immigration category, as you say your WP is expiring, unless your employer has extended their sponsorship it would be correct that you cannot extend your leave via a FLR application.
Which immigration category will you be applying for ILR under? 10 year Long Residence or the 5 year Economic Migrant route?
If applying under the long residence route see...
http://www.ukba.homeoffice.gov.uk/sitec ... iew=Binary
For applications made on or after 9 July 2012 an applicant applying for an extension of stay or indefinite leave to remain (ILR) on the basis of long residence must not be in breach of the Immigration Rules. However, for the purposes of the application a period of overstaying of no more than 28 days on the date of application will be disregarded.
This would mean, technically you could do your LITUK test (& pass it) and submit the application via post soon after, but before 28 days of your current LTR expiry date. This is a suggestion based on your current circumstances, due to the fact that you have unfortunately not looked at the requirements for ILR early enough to enable an in-time application.
