I'm thoroughly confused by the question.
First, the valve core is made of brass and plastic.
Yes, they sometimes stick, but a valve core tool is enough to unstick them. Besides the only time this is a problem is if the valve is being reused - something hardly anyone does. It is usually replaced.
But if you are talking about a valve core leaking - those sealing parts are made of plastic and while they can get damaged by dirt and not seal, lubing them doesn't do anything of value.
So I'm confused. What is the problem again? The shop couldn't remove the core?
I'm thinking they were afraid they might break the valve and have to replace it free of charge - and (I'm guessing here), since the TPMS sensor is part of the valve assembly, that would be expensive.
I vote for not worrying about it. If the valve itself works, and it doesn't need to be taken out, there isn't a problem. I would be very careful putting anything in the valve core as it might dissolve the plastic parts.
Oh and that "valve lube"? I think that is assembly lube for ENGINE valves. Not tire valves.