No you don’t have to drop below 200 to achieve your goal of 5*10.
I have 2 ideas that may help you.
Do 5 sets with something that you can easily accomplish. Say you can get 5 sets of 7 without failure. If this is so then start off with 5 sets of 5. Do this 2x per week. Next week do 5 sets of 6, 2x per week. Each following week you add a rep to each set. Within a month and a half or so, you’ll be able to accomplish 5 sets of 10.
My reasoning for starting with 55 (if you can do 57 is because this is a progressive type routine and you don’t want to top out before you reach your goal. If you start at the 57 and next week try 58 you might get it, might not…but probably won’t get to the week after of 59. By starting a little lower, you’ll be able to easily handle the first 3 weeks as your body gets used to the routine and by the 4th, 5th week etc. you’ll be challenging yourself, but you will be able to make the jumps easier as your body is getting used to adapting to the added volume.
Weeks (Assuming you can nail 5 sets of 5 reps)
5 sets of 10 is 50 reps. How long does it take to complete 5 sets?? 10 min? 15 min?
How long do you want to be able to complete 5 sets (assuming each set is 10 reps each)? 10 min?
For argument sake, let’s say that you want to complete your 5*10 in 10 minutes. Right now you can’t. The suggestion is that you break it into 20 minutes. There are many ways to accomplish the 50 reps in 20 minutes right? You can do 1 rep every 24 seconds, or 5 every 2 minutes or whatever. The goal here is to shed time off of the 20 minutes in the following workouts. If you do pullups 2x per week you can shed 1 minute off of the 20 each workout and in 6 weeks you’ll be down to 10 minutes for the completion of 50 reps.
1a 20 minutes for 50 pullups
Does this make sense?
I hope my ideas helped. Good luck reaching your 5*10 being over 200lbs.