Great question. You do risk running into some friction, but hopefully not as much as you’re anticipating. A few thoughts:

-Letting them know “I’ll start work when you pay the deposit” sends a signal that you’re serious. The lead contact — a front-line departmental manager usually — is eager to get work going because they have problems they need your help solving. If it’s important to them, then they’ll communicate to their colleagues in procurement or billing that it’s time sensitive.
-They may have a stated policy about paying “freelancers” but they almost certainly have a mechanism to pay other people and organizations quickly. Deposits via purchase orders are common for any contractor that has supplies, for example. If their offices have ever been painted, they paid a deposit to the painting contractor.
-Many front-line managers (or their immediate bosses) have a certain amount of purchasing authority via credit card. If so, Paypal is a quick way to get you paid. The fees might be a bite, but to secure the deposit you might feel the fee is worth it. If you’re using a subscription invoicing software like Quickbooks, you might have the option to receive payment via credit card already.
-On one occasion I did do something in-between for a project with a tight timeline. When the lead contact assured me the invoice for the deposit had been routed for payment, I started work even though the check hadn’t gotten to me yet. That worked out okay. It’s a risk, but if a couple weeks in the payment still hadn’t arrived, I’d be able to say “We agreed . . .” and have a rationale for pausing work.

Practically, though, it doesn’t often come to that.

Overall — and this ties into the articles about raising rates also — don’t assume you’re asking too much. I have yet to meet the truly accomplished and experienced freelancer who doesn’t have more sway with their clients than they think.