It's a large university, and this is apparently how the finance department pays all their bills, not negotiable.
The credit card processing fees are delegated by credit card companies (visa, MC, amex, etc) and are also non negotiable, it's just the cost of doing business with credit card companies. The baseline fees are for when a physical card is used at a POS (ie they are able to verify the card is physically present). When you order online, or take payment over the phone and the card is not physically present there is an additional fee that is charged. This seemed to be the case everywhere I checked.
Since the fees are unavoidable I bit the bullet and went with stripe. I based my decision entirely on what option was the cheapest for this single instance since I don't plan to ever be paid in CC except on very rare occasions. They charge 2.9% base, plus 0.5% if it's a manual entry (physical card not present), plus $0.30/transaction. I was able to set up a company account and get verified and process the transaction in under an hour. Takes about a week for the funds to hit my checking account (less the fees).
Now that the account is set up I can take payment with a credit card in a matter of minutes, it will just cost 3.4%. We are planning to amend our proposal template to include a 4% surcharge on payments with credit cards to cover the added fee.