I don't think this is a case that any PEO will be able to decide on within 24 hours (same day). Most likely, an in-person application will be deferred. Once deferred, it is anyone's guess!!
So, you are looking at a (likely) scenario where you are stuck in the UK for god knows how may days or months .... while you are absent from your project. Given what you have posted, it doesn't appear that your employer will grant you leave during this crucial period. and, given the reputation that you have seem to have created, you will not wish to fall foul for any reason.
With that in mind, if I were you, I would go for an extension (easy, in-person, same day possibly, absences of no relevance) ... and worry about settlement when the project is over, I am basking in the glory, and can demand enough leave (or some sort of breather, less stressful short term UK assignment) from employer to be in UK, and prepare for and sort the settlement issues without undue distraction. Hey, it might cost a lot of money .... but your situation isn't as normal as what normal is.
My personal opinion .....
Sorry, but have no clue about any solicitor ... competent or otherwise!
