If you start studying in the first week of May you will have 31 weeks left until the December JLPT. Let’s say you want to finish a month early so you can use the last 4 weeks to review and do some JLPT practice tests. That leaves you with 27 weeks.
If you want to finish all 12 chapters of Genki 1 in that time you will have 15 days per chapter. I needed way more time than that but I think this is totally achievable if you study more consistently than I did. 
You wrote that you want to improve your grammar so I’m assuming that you already know some basics. Depending on your knowledge level, you might be able to skip or breeze through some easy stuff that you know already and save some extra time.
Even though Genki 1 does not cover all the N5 grammar and vocabulary, it’s enough to pass the test comfortably in my experience. It does not teach all N5 kanji either but based on your WK level you seem to be covered already in this regard.
That said, you should practice listening as much as possible. Genki provides some listening material but if you don’t do any listening outside of that, you might struggle when taking the test. I can recommend The Ultimate Additional Japanese Resources List.
Don’t stress about rushing it though. It’s not a race. 
You will have roughly 2 months until the December JLPT registration period starts. My recommendation is to simply start studying, see if you even like Genki and find out how far 2 months get you. When it’s time to apply for the test you can decide whether you want to take the December test or wait until 2024.