How to improve at low 6 digit rank (~200-100k)?

I think you should keep doing what you're doing but, try fcing a lot more 4.2 - 4.7 star maps as well. (i do the same thing). Once you start getting to 120k progress gets very hard and you need to get consistently 120 - 130+ pp plays if you want to rank down considerably fast. Try to play (not fc) more 5 stars and get used to higher ar's if you haven't.

This helped me to get from 200k to 103k in 40 days with a lot of hours so I don't think it'd be possible to reach a 5 digit with your time frame but this could help.

Profile: https://osu.ppy.sh/users/14402013

(If you want the easy way just get sotarks, log off now and monstrata's maps and fc them)

Like what @Shuckling mentioned in his answer, try FC-ing more maps in the range you can play as well.

Additionally, play some maps that are 5~6 stars that are at least playable to you.
This is just for you to get familiarized with what the higher-ranked maps are typically like, and for you to adjust accordingly to it.
Do not quit half-way through even if your progress is bad.Use No-fail if you need to.
Use offline-mode or edit the map if you don't want it to affect your rank

Additionally, you can edit maps as well to make it a little bit more challenging for you. You just have to do simple edits on the Circle Size, or Approach Rate, etc.
Plus it does not count towards the online leaderboard too if you are training, if that matters :V

Plus, an obvious one, practice. But practice smartly.
Try to find out why you are bad at certain areas and adjust accordingly.
Be a little bit more conscious on how you are tapping and moving around the cursor, it can help you to find out where you can improve at.

Also, what I personally do is this: I search for players near my rank's range, and then try to 'one-up' their Best Performance maps.

I'm a mouse-only player, currently at #94k but peak was #89k. I hit the same wall. For me the biggest thing that helped me was taking about a month off and then derusting hard for 3 days. On the third day I would be able to get about 15-20 new top 100s somewhat effortlessly (relatively). I've done this a few times and it works best for me. When I life the game for too long I get sick of it and tilt a lot more often and the break helps with that.

Also, I went for my first 6 stars around that rank. There are some relatively easy ones like Flamingo. I'd recommend sorting your collections by difficulty and running through your easiest 6 stars, you will probably be able to pass one (if you have a large collection of maps).

I recommend Tillerino for recommendations if you aren't sure what to play since it's based off other players who have passed what you have passed.