わけ is attached to the phrase いつも紙とペンを持っている, in other words
that/the case that i will always have a pen and paper
でもない is modifying わけ, in that
it is not really that わけ
putting the two together,
it's not the case that i will always have a pen and paper
in terms of the difference between も and は in this sentence for でもない・ではない,
from Meaning of 「という訳でもないのだが」 :
"～という訳ではない" is a common set phrase which corresponds to "That is not to say ～" or "That doesn't mean ～", referring to what was already stated. Using でもない instead of ではない adds "not in particular" or "not really" feelings to the sentence.
I think this somewhat applies here. The basic grammar is ではない but でもない sounds more natural because でも instead of では has a nuance of "even" because of the "also" aspect of も.
So to give a more colloquial translation, "It's not like I'll always have a pen and paper" as opposed to "I will not always have a pen and paper"