I believe your thinking is correct. It sounds like you have already thought this through, but I will try to quickly flesh out how I would analyze this on pen and paper.
Lets start by assuming you are correct in both your questions: 1. Phrygian#3 and Harmonic Minor are essentially the same thing. 2. Phrygian#3 is also a mode of harmonic minor.
To hash it out, lets take C Major as the 'Parent Scale': C D E F G A B C
Play C Major starting and resolving with the 3rd degree, and you have E Phrygian: E F G A B C D E
Play C Major starting and resolving on the 6th degree, and you have A Aeolian A B C D E F G A
NOW lets try and find the correlation we're looking for....
As you pointed out, raising the 7th in an Aeolian scale gives you harmonic minor. So lets raise the 7th of our A Aeolian.
A B C D E F G# A
Now, lets see what happens when we raise the third of our E Phrygian:
E F G# A B C D E
As you can see, both the Phrygian #3 and Harmonic minor contain the exact same notes. Further, Phrygian #3 also begins and ends on the 5th degree of your harmonic minor scale... In this sense, Phrygian with a raised 3rd is derived by playing the 5th mode of the harmonic minor. However, I have normally heard this referred to as 'phrygian dominant' or 'phrygian major'.
Anyway, it sounds like you have already thought this all through. This is just a version of how I would analyze scale relationships.
If anyone has any tips or suggestions on how to think about this I would love to hear it!