The final answer is:
STOREROOM CLASSIC PUZZLES
This fits, as the text at the bottom of the sign that reads "FA: 9,7,7" represents FINAL ANSWER: 9,7,7 where the 9 and two 7's represent the lengths of the words which make it up.
To solve this, first consider the two poems. These provide some clues as to what is expected of us. Most notably:
1. In poem 1, the use of "the choice, it is two" conjures up the idea of binary. This is further emphasised by its acrostic spelling 'ROOT', which is a synonym of 'BASE' and suggests we should be looking at arithmetic in some base or other - binary is base 2.
2. In poem 2, the acrostic spells 'MORSE', which suggests we should also be utilising Morse code somehow.
3. Furthermore (coupled with the hints), the talk of 'Rotations' in poem 1 suggests the use of a Caesar shift at some point (like PSE's favourite, rot-13). Plus the emphasis on the colours 'red' and 'blue' indicate we may end up using their positions in the rainbow - 1 and 5, respectively.
Turning to the top half of the diagram:
If we ignore the colours altogether (a 'crimson pisces' - or 'red herring' - perhaps!) and realise that the black line separating them should also be considered an important part of the puzzle, then we have 7 rows of 5 (binary) bits. If we treat the coloured cells as 1's and the black cells as 0's, we see the binary numbers:
01000 / 01011 / 11010 / 10010 / 10010 / 00010 / 00010
Converting these to decimal, we get:
8 / 11 / 26 / 18 / 18 / 2 / 2
Converting these to A1Z26, we now get:
We'll come back to this later...
Now for the bottom half of the diagram:
Again ignore the red-herring colours, and notice that we have 7 rows comprising a combination of 1-cells and 3-cells. Since a Morse code dash is 3 times the length of a dot, this suggests this is where the Morse comes in. The diagram now translates to:
-. / ..- / --. / --.. / .--. / ..- / -.-
These are equivalent to the letters:
We now have two strings of letters - what do we do with them?
Well, recall the talk of 'rotations' from earlier, and the fact we have not yet used our colours and their rainbow positions... If we apply rot-1 ("rot-Red", perhaps) to the first and rot-5 ("rot-Blue") to the second, our strings become:
HKZRRBB -- (rot-1) --> ILASSCC
NUGZPUK -- (rot-5) --> SZLEUZP
Using the anagram tag means these will rearrange to:
CLASSIC and PUZZLES, which conveniently fits two of the 7-letter words we expect to find in the final solution (which, recall, has format 9,7,7).
Lastly then, to find the remaining 9-letter word, note that we already had a set of 9 letters that might be 'anagrammable' - the two acrostics: ROOT and MORSE. These conveniently anagram to STOREROOM - and a storeroom of classic puzzles could very well be an appropriate room to lie beyond the PSE lounge!