What are some approaches to attack an triple key polyalphabetic cipher when the IV key and PT key are known?
I can't comment yet and have no solution, but I still wanted to post my own analysis and hope it'll get you a bit further. BTW, the site you linked to with the info about Quagmire IV is mine and yes, sorry about the ads! I haven't updated it in many, many years, so I was surprised to see it pop up :)

The first thing to do is try to find plausible candidates for letters in the keyword. I'm absolutely assuming the CT key was produced in the same way the PT keyword was, i.e. a keyed alphabet and not a completely random one. The latter case will require hill-climbing algorithms and is not in scope at this time.

From the crib I could deduce:


P and R are 3 letters apart (2 between them). Since in the regular alphabet there's only one between them, I think both are used in the keyword. PxxR. If you put the R in front, the P ends up at the back of the alphabet and this is unlikely, since that would require almost every letter after the P in a regular alphabet to have been used in the keyword.
I and X are 13 letters apart. I think the I is not part of the keyword, because if you put the I at the front somewhere the X is totally out of place somewhere in the middle and it could only be the 11th, 12th or 13th letter in the keyword and I assume the keyword is shorter.
However, the X may be part of the keyword, since the I ends up near its original location when the X is put at the front, but this requires that at least 5 letters that normally come after the I appear in the keyword, plus one extra letter for every position that X moves to the back. This makes it more likely that the X is the first or second letter in the keyword, than any other.
The M and S are 10 letters apart, where normally they are only 6 letters apart. This implies that either one of them was used in the keyword. Both are possible, since the other letter would end up reasonably close to their original positions in the alphabet. If the M was used, it is most likely one of the last letters in the keyword; if the S was used it is likely one of the first three or four.
The A and F are 24 letters apart, as opposed to the original 5 letters. From this it is highly unlikely that the A alone was used, since that would leave the F way at back of the alphabet. Either both were used in the order of FxA, with one letter between them, or the F alone was used as the second to last letter of the keyword and the A is the beginning of the padding.


Putting this all together, I believe the keyword definitely contains P??R, S or M, and F?A, not necessarily in that order. It may contain an X at the beginning as well.