This is one of those times where the SSN/EIN having become "overloaded" in its use as a unique identifier bites back. Yes, for tax purposes, an LLC need not have an EIN, but every financial institution I've dealt with for mine says "yes, you do need an EIN for us." For example, my credit union requires an EIN for all non-human-based account owners, whether or not the IRS requires one.

I am absolutely a believer in "I want to do things the way I want to do them" but in this case I believe this is an argument your wife will lose on her LLC. The only way to play is to get the EIN as requested; financial institutions are notoriously inflexible.