- A publication doesn't need to be OA to be useful for your CV or for the profession as a whole.
The standard procedure "should be" that the university, or at least the department, have a stated policy on paying fees for doctoral student publications. If they insist that you need publications for the degree then they should be willing to pay the costs of that.
Open access fees are another matter, however, since they are high. While the principle above still applies, I doubt that many places would pay the full fees for student publications, whether "independent" or not.
A typical faculty member will seek grants that provide funds for publications under the grant. Some faculty members will have funds for publication that are more broadly based and less restricted.
But the choice to go open access is yours. You understand the costs involved. If you aren't willing to ask the professor directly or indirectly then you are probably stuck with those costs.
I'd actually recommend, in your situation, that you publish in a more traditional venue. There might still be page charges, but much less than open access and with a higher likelihood that either the university will absorb the cost (a good policy) or that the journal will forgive the fees due to the fact that you are a student (independent or not). That is fairly common in publishing, or was when I was still active.
Save OA publishing for later in your career when you have the opportunity to develop funding sources for such things.