How to determine which type of transposition cipher I have?
However how do I determine which transposition cipher I would need to use? Am I right to assume this is definitely a transposition cipher also?

As you found out that the ciphertext has a similar frequency to the English letter frequency. This can lead to lots of possibilities

It can be a shift cipher, that is the generalization of the Caesar cipher $$c = m + k \bmod 26.$$ In the Caesar cipher, the $k=3$ is fixed. There are at most 26 candidates to test, not so far.

It can be a permutation cipher, that is there is a permutation $\pi$ on the alphabet that permutes the letters. The 1000 years old frequency attack work on this. You need to swap the top three letters and try to reveal some other. Since you have a long message this should be working very well. Actually, the substitution cipher is also a permutation cipher.

It can be any of the transposition ciphers since they are keeping the letter frequency, too.

Rail fence cipher
Try some possible rail lengths. It is a very weak cipher. The below is the Rail Fence histogram with rail size 3. This is exactly the same as Moby Dick's frequency since the letters are not permuted, just the positions changed. The same applies other transposition ciphers since they just transpose.






Scytale
The cylinder size cannot be too long. try some possible sizes.

Route cipher
Row size and route to read the letters are required. Don't expect the road to be too complex. Set a row size and test for well-known routes. spiral inwards.outwards, clockwise/anticlockwise, starting from the top right/top left, bottom right/ bottom left, etc.

Columnar transposition


 Guess the column size and look for digrams to determines the permutation on the columns.


Double transposition
This is hard to attack, it takes the result of the column transposition and applies another transposition. Still, we have computers, and once can test all values up to some limit and check for the string in the decrypted text. Linux's strings command can be helpful to distinguish those.

Myszkowski transposition
Just a modification of the Columnar transposition that deals with the reoccurrence of a letter in the key. Increases the key space a little bit.

Disrupted transposition

Grilles
This is one of the hardest ones in this series since the attacker needs to form the grille. Gaines book's chapter is advised. It has some variants.


We don't expect combinations since the combinations can hide the frequency.

Vigenère cipher
It is not expected to be Vigenère cipher, however, the Kasiski test can be used to test.


Online Tools

Crypto Tools 2
dCode.xyz
boxentriq


I also did;

The light gray is from the Moby Dick text, the medium grey is your text.
In a bar, a light grey on the top means that moby dick has a higher frequency, a medium grey on the top means that your text higher frequency, a sample Python code, and Moby Dick is here.
From here we can simply say that it cannot be Caesar or Shift Cipher since the frequency is very close to the real English frequency!.