"and then use the result as their master password on LastPass"
The proposed "algorithm" is in essence just a fancy key expansion.
As you surmised, it is security through obscurity. If one knew that I was using a base64 encoding of a plaintext crunched through (say) Bifid, he would be able to brute force the database much faster: the plaintext would probably be no more than four or five words, giving somewhere between 50 and 70 bits of information, not as if it was a real 100-bit sequence.
On the other hand, 50 to 70 bits of information is pretty good for a password (see mandatory xkcd link). So if you started with a secure choice, you'd stay secure.
On the gripping hand, good password managers allow for any length of master password. So I could choose a lengthier text of my choice, and that could be both easier to regenerate and better security. I can't run Bifid with pencil and paper every time I need to unlock my password manager.
And if one chose a single word, insecurely, trusting in the magic of
base64 to enhance its hardness? If that became known, a password database could be blown open in a few minutes. So, depending on who's given to, it might even be a very bad suggestion.
A colleague told me that, apparently, this kind of - not "fallacy", let's say "inefficiency" - might be called Feynman's Yellow Paint. The story goes that a painter claimed to be able to get yellow paint by mixing red and white, while Feynman held this was not possible. They proceeded with the experiment, and the white-red mixture turned pink -- and stayed pink. So the painter commented, "Usually I added some yellow, and then it turned out fine".
The analogy here is that to get a strong password from a base64'ed ciphered scheme, you need to start with a strong password. But if you have a strong password, then you do not need the cipher and encoding superstructure.