Amamo or Iki in July

The only time we can go to Japan is mud July, for 2 weeks. Our trip ends in Osaka on 25th July. I can't decide which is better for some nice beach days from 25th to 29th July. Would Amamo be 5 days of rain and wind? If that happens in Iki,being based in Fukuoka means we could at least do something else instead.

I can only go on some assumptions and inferences of your post - first, I presume you really mean Amami-Oshima? And from what you write, your primary interest is in swimming in the ocean?
If so, Amami has a vastly longer coastline, with numerous places to swim. Iki as far as I know, really just has one - Tsutsumi. Either way, please note that none of them are large places or very developed and with lifeguards.
The only practical way to get to Amami is to fly - fortunately, there is a flight by JAL, and you can use its Explorer fare for 11,000 yen one way if a foreign tourist. There are few flights however, which makes your schedule tighter. For timing, you mentioned mid-July, and normally the rainy season has ended by that point, though it can get very hot and humid. Amami has much more of a semi-tropical atmosphere though compared to Iki.
Iki is much closer (about 20 km from Kyushu), and the ferry ride can be nice to get there (it has an airport as well, but the short hop flights are quite expensive, and I don't know of any discounted ticket you can use). It has some beautiful coastline, though more scenic and rugged - not for swimming.
You mentioned 4 days though - that is way overkill for Iki, and probably for Amami as well, unless you just want to chill there. If you're going to do all those days, then I suggest you try Okinawa. Or another option is Yakushima if up for the gorgeous forest hiking. There are flights from Fukuoka Airport as well, though no discounted one for Yakushima. Okinawa has loads to see and do, and some very nice beaches. Yakushima has a few beaches, and unique primordial scenery.

Thank you, your information is very useful.
I did think of Okinawa, but can't find anything budget friendly on booking.com. Most cheaper properties are right up in the north. We won't be driving so that means an expensive taxi ride and more time just travelling.
I thought Amami Oshima ( i mean the place called Amami on the map, south of Okinawa, there is I think, an Island of the same name nearer to Tokyo too) would be quieter. I have got some reasonably priced accommodation there. However, I am worried that weather could prevent us flying there, or worse still, flying back to Tokyo to get our flight home.

I booked a place in Hakata too, because I thought if the weather is bad, there are other things to do apart from beaches. of Iki. I would much rather chill and have a nice beach and a nice swim in Amami Oshima though, but would be sad if we spent 3 days in the rain there.

I noticed your post while browsing through the forum for the first time. I'm not sure if you'll still be going to Japan, but I'm replying since I have relevant information for you (or others with similar questions in the future). I can't comment on Amami since I've never been there, but I actually live on Iki. Iki has several wonderful beaches for swimming, both on the mainland and on easily-accessible outlying islands. The most spectacular beach is on an uninhabited island to the north called Tatsunoshima. You could easily spend a whole day there as there is a huge, beautiful lagoon as well as nice hiking. I will also say, as someone who's lived in Okinawa and Iki, that while Okinawa has far more to do, Iki is safer for swimming in the summer as there is no risk of box jellyfish. Iki also has many interesting sightseeing spots such as monkey rock, kojima shrine, and harunotsuji ruins. I think you could fill three days with fun activities and enjoy delicious food (sashimi, champon, michelin recommended pizza, and some of the highest ranked beef in Japan). You can access Iki via plane, ferry, or jetfoil from Nagasaki, Karatsu, or Fukuoka, making it a great place to hop around northern Kyushu. You could also island hop from Iki to Tsushima. If you end up coming to Japan next week, I wish you a very enjoyable trip!