You get a lot out of a PhD, or at-least I did out of mine - in particular analytical, research and writing skills. Best of luck.
As for flying training - well integrated course providers don't really care so long as you are up to the course, and of-course can pay the fees. So, I doubt that will matter.
And at the end, what are you qualified to do? - well you can either be an engineering postdoc, or you can be a junior professional pilot. Both interesting and worthwhile jobs, also both jobs that are quite difficult to get. But, you've got double the odds of being employable.
And after that? Well, if you climb the pilot career ladder and start to look at technical or management roles, the skills learned in your PhD will do you a lot of favours. Or if you go back into academe, you'll rapidly discover that there are lamentably few aerospace academics who really have a clue about aircraft operations - so you'd have a big advantage there too.
So, a slow and expensive route in - but a route into some really quite exciting opportunities for you. Plus there are a few jobs out there (and generally not enough people to fill them) for people with a combination of professional flying, and academic/aeronautical qualifications: at places like NTPS, ETPS, NFLC, BFSL....
One thought however; you'll almost certainly stand out as a pretty exceptional candidate anyhow, and I doubt that all that extra money spent on Integrated will really add much. If I were you, I'd save a few tens of thousands of pounds and go modular.
G