Is it Islamically forbidden to photograph a girl one intends to propose to with a phone without her knowledge, then delete the photo, or does this fall under the permissibility of repeated looking for the purpose of achieving the benefit of marriage?

We have not found any statements from scholars regarding secretly photographing a woman for the purpose of engagement. Permission has been granted to look at a woman when there is a need to propose to her, and to repeat the gaze if the purpose is not achieved with a single look. If you need to look at her again to ascertain her condition and appearance, you can request to do so. As for proceeding to photograph her secretly, it is preferable to avoid it due to the significant difference of opinion concerning the ruling on photographic imaging of living beings, and because the matter is graver when it pertains to photographing women, in addition to the absence of scholarly permission for such a situation.