Which train company and route is the best for this ?
It depends what you're after. Speed-wise, then a direct TPE service. The first class seating on the TPEx IEPs is the same as on LNER's.
But the TPEx first class isn't anything to write home about: snacks and tea/coffee. That said, if your journey on LNER is under 70 minutes (and York-Newcastle is 60 minutes) then you only get snacks too*, though you might get a beer/wine as well as a tea/coffee depending on time of day.
If you want to mix it up a bit, then perhaps get the Scarborough train then change at York. Other option would be up the WCML, but 1st class on a 156 on the Tyne Valley is definitely non-existent.
I've not looked from Liverpool, but from Manchester Airport (MIA) it is sometimes possible to force journey planners to book MIA-Newcastle via the Hope Valley and Doncaster, even on advance tickets. I've gone that way before on Airport Advances from MIA to Newcastle. I'm not sure if it would work from Liverpool though.
(* Those are the rules. My experience is that northbound from York the catering crews stick to the rules, as the crews normally change at Newcastle they start to pack up after Doncaster. But southbound from Newcastle they generally just serve everyone the full menu, or at least the sandwiches, unless you get a particularly pedantic first class host).
TPEx were also meant to be offering Carluccios pastries and pies on weekdays but I have never ever seen these onboard.
Carluccio pastries, yes, and very nice they are too (though unsure if they still do them after Carluccio went into administration). Never seen a pie though.