Have to start with the disclaimer -
In the United States it is absolutely "illegal" to alter the horsepower of a stock motor such that it's output is different than the emissions sticker on the motor.
Personally, not living in the U.S. I am not governed by those rules. However, if "you" live in the U.S. any advice I may give is for informational purposes only
To start with Merc used two different powerheads for their 9.9 horse, 4 stroke models. One was shared with 8 horse (serials starting 0Hxxxxx) and the other with the 15 (serials starting 0G59xxxx, then skipping the 0H series and continuing on). The smaller block was just barely over 14 cubic inches so a carb swap on that one will do nothing for you (and actually detract from the performance of the motor).
If you have the model with the larger block (19.7 cubic inches) it could be modified - and O your research has shown, the only difference I can see is the carb.
Having never attempted to convert a 4 stroke model, I can not tell you if you could simply change the jets or if the throat on the carb body is also different. It could end up meaning a total replacement carb (at a list price of about 210 bucks).
Now, don't want to "rain on your parade", but if you find a 9.9 is wayyyyyyy underpowered you will not be much more impressed with the 15 horse. Because they are essentially the same motor the only place you get the "extra horses" is at the extreme top end of the throttle.
So what a increase to 15 WILL NOT do for you -
it will not get you out of the hole quicker (torque curve is identical)
it will not get you on plane faster (ditto)
it will not get you "gobs" more speed at the top end (a 50% increase in horsepower translates into 15% more speed, and only at the top end)
So now if your top speed is say 18 mph, with a 15 horse you will barely top 20 mph.
(I have done dozens of "non-scientific" tests between 9.9 and 15 horse performance (as well as 6 to 8, 6 to 9.9 etc) because I have been asked and done more than two dozen "conversions" for guys - the results of the conversions "disappointment and a thinner wallet" - my advice always is, replace the motor with a larger displacement model if your rig can handle it)
What it "will do" -
use more gas all the time with no noticable performance increase (I have yet to actually find someone who can tell the difference between say 23 and 25 mph unless they were staring at the speedometer or gps)
Your boat is rated for a 20 horse so if you really want a performance improvement it would be worth going from the 9.9 to Merc's new 20 horse. It is only a few pounds heavier than your 9.9 and uses a 21.4 cubic inch powerhead - and when you are looking for "more power" (which most times means "more torque") there is "no replacement for displacement".