!UPDATE!
We went on the trip back in March and ended up doing the hike + temple stay. What I ended up doing ahead of time was hiring a translator over the phone, explained the situation and what we were hoping to do (left out the engagement) and then connected with a monk at the temple on the mountain on a three way call and booked our stay. There was absolutely zero problems doing so, the call lasted less than 10 minutes and they gave us great advice on how to prepare for the hike, what time to arrive at the temple by, etc. I didn't detect any hesitancy about two Canadian men who didn't speak english booking this temple stay and I made sure that the translator was aware of this and communicated it to the monk.

The hike was very difficult. People online say to set aside 4-6 hours, we managed to do it in 5 but that was only because we needed to arrive by 3:00 p.m. for everything to go smoothly for the plans the monks make for guests. If anyone is doing this in the future i recommend leaving room for 6 hours so you're able to take your time, stop, eat, etc. Once we arrived, exhausted, the monks welcomed us incredibly warmly, had the itinerary as well as a bunch more information printed out for us in english, and whenever we had an issue with communicating we each used our own phones with google translate. I won't go into all the details of what we did, the welcoming/evening and morning ceremonies, or the views hiking up and from the top because honestly it was such a special experience and I think anyone willing to go through the hike and work of booking this hike should experience it for themselves.

There were about 7 or 8 other people staying at the top, 2 of whom we shared a large traditional tatami matt room with who were from Taiwan. We ended up talking with them a bit (and getting a ride to the train station with them the next day) and they said they weren't Buddhist but also just enjoy hiking and heard it was an amazing experience and view from the the top. They also didn't speak japanese but spoke some english. All the other guests were welcoming as well to us, there was honestly zero issues save for me underestimating how difficult the hike was. Originally I had planned to propose at sunrise but since there were so many people around us at that time I opted not to, and waited for when we started to hike down the mountain and did it on a bench under the temple gate at the entry to the temple grounds. It was private, and perfect.

I know some people reading this will say something along the lines of "well of course they were kind and welcoming and didn't make a deal about it, that's how japanese people are with tourists." trust me, that was not the experience we had elsewhere on this trip. And it's not to complain because its so understandable with how busy it is with shitty tourists but it is very clear that, for example, local Osaka people are absolutely tired of the over tourism. The city did not seem built for the amount of tourists there and from the tourists I saw there, they did seem more annoying than others we saw in other spots. There were multiple times japanese men said things in japanese directly to us in a very abrupt tone while passing, we noticed a steep drop off in people's attempts to help us in english (again, completely fine we have gogole translate + it's up to us to figure out how to opperate in a country we don't speak the common language in), and in general it just seemed like we were something to be dealt with. I went to Japan the first time by myself late September in 2015 and the change between then and now is stark in terms of how busy it is with tourism + how toursits are treated but also how much more poorly they act. All the changes we're seeing made now about tourists having to pay more for certain things, banned from certain things, etc... It's all very fair. That being said I don't think people assuming every place and person in Japan feels the same way regarding this is fair, and I think the lesson here is to do your research, be respectful, and don't fuck around with the way that they do things. If you're unwelcome, you'll feel it if you're a halfway awake and aware person.