An answer from the paddle board side. I've seen paddle boards up to 400lb among the casually available, generally among the longer (and wider) of the selections, like 12-14 feet or more.
With some looking online, I found one that is rated to 450, another for 430, intended for tandem paddle boarding. One more was rated to 700. There are also group paddle boards, the first link mentions one rated to 1050. Note that this would be very large and likely difficult to maneuver. Searches for tandem or family sized paddle boards may unearth others.
On the personal side, I took a paddle boarding lesson some months ago and tried several boards... looking at the average capacities vs lengths now, at least some of the boards I tried must have had official capacities considerably under (100lbs or more under) my actual weight. I had no real trouble with any of them, though I did prefer the longer and wider among the boards. It may be relevant that I didn't stand up to paddle board, I was kneeling and sitting on each one I tried - I mention this because difficulty balancing is one of the cited problems with overloading a paddle board, though my own balance is poor enough I can't say if that was the reason.
It may also be worth noting that weight and weight capacities were not discussed when signing up for lessons, contrasted to kayaking which both noted some restrictions when signing up and considered capacity when choosing a kayak. Also, I had quite severe problems balancing and maneuvering while kayaking (probably I was close to or a bit over the capacity), while the paddle boards were quite workable and stable enough I would have had no concerns about adding weight (in the form of gear) even though I must have been, again, nearly at to slightly above intended capacity.
The reason for this somewhat long-winded anecdote is, you may be able to look for lessons or rentals on boards even somewhat below the recommended weight capacity, as long as you can account for the lesser stability (in my case, by sitting or kneeling instead of standing). This might work to see if you would even be interested, and you can look for one with the right weight capacity after you know it will be worth it for you.