Some questions...

I'm with you in that it didn't look like near enough money. Someone with knowledge of money would have to say whether or not they bad larger bills than $100 at this time period.

1. They knew she'd take the money; wouldn't you?
2. You weren't watching close enough. Look carefully, you can see (as she shoves the last wad in) that her purse is full of cash, as she'd been 'tapping' the drawer all day long. In reality, this is ridiculous, as 100k in cash did not flow through a teller's window back then (nor does it now, except in very few places). And when she took her cash-box to the 'vault', what was in it? Another 100k? Really?! The real puzzler is, why would she tell the guy that works with her and hits her up for a movie to 'call me'? He'll be there the next day! How real is that? Guy is desperate -- but Lee IS hot!

I had many of the same questions. Did Lee Remick really embezzle the money? And for that matter, did the bank leave that kind of spare change, to the tune of a hundred grand, lying around? Wouldn’t they notice the missing cash before she left the building? I recall an earlier scene in which a bank bureaucrat said the bank vetoed the notion that she would steal real money. So did they change their minds and not bother to tell the audience? This part always rang a little untrue to me.

I'll use the spoiler card from here on
The bank knew all about the plot. They wanted to use phoney money at first, but Kelly thought Lynch would check it out before releasing Toby. FBI agent Ripley 'assured' the bank manager that they would cover any losses. The concern I had was the amount of currency needed to reach the $100K. Very doubtful that any super large bills ($1000 or $500) would come through her window -- they were still out there in '61 -- so even with nothing but $100s it would take 1000 bills, but we only see $20s in her wad. As for the end-of-shift count, since the manager was in on the plan, he would have just gone through the motions so the other workers wouldn't suspect anything.
Hope that clears things up for you.