That is correct: $P!$ (which I call $P\#$) is so huge compared with $P$ that you can effectively ignore what happens below $P$. The estimate in my previous comment for $Q$ (the number of primes below $\sqrt{P\#}$) is that it is approximately $\dfrac{2e^{P/2}}P$.

In your initial post, you said that you wanted the primes between $P$ and $\sqrt{P\#}$. Now you are asking for the range from $P$ to $P\#$. I'll stick to the square root case, but if you want it without the square root that can be done too.

You don't need to estimate the sum of reciprocals of the primes in that range by taking an "average" prime in that range and multiplying by the number of primes. That would not be accurate enough to give you a valid estimate. My previous post gave a better estimate for that sum of reciprocals, namely that it is approximately $\log P.$