You will make things easier for someone looking to compromise your password irrespective of that second factor. You're under the impression that any second factor will be more secure than the first. Think about that for a moment.
Imagine you put your trust in Acme Dual Factor company. You trust that they will be your "overseer" in the sense that you have one key, and they have the other. Chicken or the egg concept. If one was that secure, you wouldn't need the other.
What I have done to manage complex passwords each different across many systems is, I have made myself a mental framework to remember them. It's based on the "something you are, something you have, and something you know" concept.
Something I have: (A Gmail account) Gm@!l
Something I know: (password I choose) f!d0
Something I am: (from New York so I will use) y@nkees
"It would take a desktop PC about 3 quintillion years" https://howsecureismypassword.net/
Let's do this again.
Something I have: my bank login (C!t!b@nk)
Something I know: my phone number ... shifted - @)@%%%(*&^
Something I am: from New York
Not my bank, but Gmail:
These are things I can NEVER forget. My phone number holding shift. Where I am from, and what is it I have. There are many mechanisms to commit to memory in order to remember ultra strong passwords. My Truecrypt password is 26 characters, my sign in, 21. Completely DIFFERENT passwords. I remember them all because I created my framework. I back them up using Keepass for safekeeping.
Now, someone will come along and say: "the vulnerability is in your framework" and I will point out, the odds of compromising ANY of those bits of data via say "social recon" is absurd.