Either and or could and does install baseboards. Depends on who you have contracted to install them. They can go in months later, does not matter. Is the real reason that the floor guys has it in his contract as does the trim guy and you now want a credit from the flooring guy for not doing the base?

Flooring guy is now insisting on doing the base but didnt have it in his original number. Giving me an outrageous number for installation and telling me the wood will buckle if we dont let him put it in.

Play them off each other for a better price.

" telling me the wood will buckle if we dont let him put it in. "

He is BS'ing you. Do what you want but I would not have him install it. Of course of there is an issue he will most likely never warranty his work.

Right on. Floor installers don't do base often. Shoe or quarter round, yes.

It is fine to install baseboard after floor install I hate 1/4 round and if the flooring is done right there is no need for it. The flooring is full of BS.

I'd be taking a second look at his methodology for installing the floor. And his qualifications to do so. There's a difference in a "floor guy" and a Flooring Professional. What industry trade groups do they belong to? What industry training events do they attend? What about manufacturers workshops? Finishing workshops?