I think you are going about it somewhat backwards. The first step is to determine where the money will actually come from: to request funds from whatever sources, internal or external, you think might be willing to pay. If your request is approved, then that entity will tell you how to go about actually processing the payment. Possibly by invoice, possibly by reimbursement, possibly by someone with a university credit card filling out a web form for you.
If you end up having the cost split between several sources, you'll have to work with them together to figure out the payment details. If it's multiple funds within the university, you'll probably have somebody make one payment and then reallocate funds internally. If you end up with both internal and external funds, it could be more complicated - expect to spend some time making phone calls and emails to get everyone on the same page.
Some possible directions to look for funding:
Some universities have funds specifically set aside for open access charges. It might be run through your library, or a research or grants office, so ask those people. (But it sounds like maybe you have already checked this.)
There might be internal programs where you can apply for funds for general research expenses.
Your chair / dean might have access to various "slush funds" that can be used to support research, e.g. from outside donors. If they agree to use it, it will probably be as a one-time thing, so you'll want to make a more solid plan for next time.
You mentioned that you have a federal grant, but that publication charges are not allowable. Double-check that, including whether other parts of the budget can be shifted around. But as another possibility, grants almost invariably contain a rather large "indirect costs" (IDC) item, which is basically unrestricted money straight into the university's pockets. Ask if the publication charges can be paid from your grant's IDC funds.
(You mentioned your university doesn't have a grants office to help you with this, which sounds like a recipe for disaster. More serious research institutions have lots of dedicated staff working very hard to comply with the government's elaborate regulations in order to maintain their eligibility to receive grants. And even then, there are horror stories about things getting screwed up. Tell your university that if they want to have federal grants as an income stream - and believe me, they do - then they have got to provide appropriate support.)
If you can't get funds lined up before the bill is due, then ultimately your choice is to either pay it yourself (personal credit card is likely to be the simplest way), or else to withdraw the paper. If you pay it yourself, you can still try to seek reimbursement, but don't count on it - you'll no longer have much leverage. I would treat that money as gone from your pocket, and a pleasant surprise if you manage to get any of it reimbursed - and be asking yourself whether you want to be working for an institution that makes you pay for your research out of your own salary.
Don't have the publisher invoice the university unless the university has already agreed to pay; i.e. unless and until the expense has been approved by someone with authority to do so. Otherwise, the best outcome is something like them docking your paycheck for the amount. The worst is getting fired for financial misconduct. Universities take their internal financial rules very seriously.
If you have no institutional or outside funds available, you can ask the publisher for a fee waiver. However, these are usually intended mainly for authors from developing countries, etc, so if you are from an otherwise wealthy institution or country, your odds of getting a waiver are not as good. But they might at least offer you a discount.
Next time, get all this sorted out before you decide to submit to a journal with APCs. And if you can't get funds lined up in advance, you should seriously consider submitting to some other journal that won't cost you money, even if it's not your top choice as far as prestige or fit. (But, make a stink to your superiors about how the university is passing up an opportunity for high-visibility publications.)