An Institutional Access category has to be bought directly. A membership entry cannot cover it, and the refusal is enforced on the server as well as in the registration window, so there is no way to spend an entry on one by accident.
What happens when you pick one
If you had already selected Use Membership Entry and you then choose an Institutional Access category, the window puts the payment choice back to paying and shows this message: "Institutional Access must be purchased directly and cannot use membership entries".
The switch is visible rather than silent. The selected option moves back to payment and the button relabels itself Proceed to Payment. The message clears after four seconds, while the payment choice stays where it was moved to, so read the button before you continue.
If you choose the category first and then try to select Use Membership Entry, nothing changes at all. The same message appears and the choice stays on payment.
Your entry allowance is not touched
Nothing is deducted. Even if a request to use an entry reached the server it is refused there: "Institutional Access cannot be purchased using membership entries. Please use payment option." What has changed is that you are now on the paid path, so check the amount before you pay.
The choice is not always offered
The option to use an entry appears only when you hold a membership that still has entries left and the competition is not free. Weekly and Monthly memberships include no entries at all, so members on those plans never see the option anywhere. See The membership entry option is missing.
Anyone can see an Institutional Access category
These categories are listed publicly on the competition, alongside Student and Professional. Seeing one does not mean your institution holds Institutional Access, and choosing one does not enrol you in it.
Institutional credits are a different route
Institutional credits are never spent from the competition registration window. A student submits through their institution's own flow, and two things have to be true: you are a member of the organization, and the competition has been assigned to your organization. If either is not true you are told "User is not a member of any organization." or "This competition has not been assigned to your organization."
See Institutional entry error messages and Registering with Institutional Access.