Last year, I received a grant to do undergraduate research with a professor at an institution different from my home institution. I worked on a project with them, essentially full time, from September to December 2022, and corresponded with them about preparation for the project the previous summer. I knew them prior to the project, having met them before I started my undergrad and having taken a class with them in second year, and doing quite well.
The project continued part time in January, although the grant had ended, and continued through to February, when the prof informed me they had started to get busy with coursework and would have to put the project on hold till the summer. I agree to get in touch with them in the summer. However, I ended up being busier that summer than I was during the semester, and didn't get in touch with them about resuming the project until late August, when they informed me they were busy until near the end of September 2023.
We got in touch again around then about picking up the project, and set a time to meet virtually the following week, but at the last minute the prof had to reschedule to the following week. The following week, we discussed rescheduling, but the professor was too busy to meet in person that week as well, and asked if I would be okay to just discuss the project over email, rather than meeting virtually.
At the time, I didn't have time to respond, so I put off replying to the message for several days, figuring that the prof was likely busy as well and wouldn't mind. However, other things I had to do came up, and it's now been several weeks since I received their last message.
However, the problem is that I need to ask this professor about writing a letter of reference for grad school. I have two other references that I believe are strong, but this prof is absolutely my next strongest reference. Anything else would be a "student got a good mark" letter, and even those it may be difficult for me to get, as I haven't maintained relationships with any profs other than the three I've mentioned here.
Beyond my usual anxiety that the professors I work with dislike me, I don't believe that this professor is unsatisfied with the work we did together. Although it wasn't enough to publish, it could become that in another couple of months of work, and I am genuinely interested in continuing the project.
I don't want to contact the professor out of the blue after several weeks of not replying to their email about the project asking for a letter of reference, but my window of contacting them is getting small (I want to give at least a month of notice, and most of my applications are due in earlyish December).
My go-to would be putting together a summary of what we've done on the project as well as a collection of things left to do and my questions about those things (considering how long its been since we worked on it, I think a recap would be welcome), sending them to the prof with an apology for the delay, and then a week or so later asking for a reference, but I'm concerned that any attempt to do so would come off as rude (or make it look like I don't care about the project).
Is this a no-win scenario for me? Should I give up on asking this person for a reference? Or am I making this out to be much worse than it is? How should I proceed in this situation?
TL;DR: Prof and I started a project last year, we put it on hold, and then played email tag for several months about starting the project again, and it's now been several weeks since I replied to an email from them about the project but need to ask for a letter of reference. What's the best way to handle this situation?