I agree with Marda on why it might be so fragile - it could be the logo - we had an instance where I washed an item of clothing and the logo kind of looked warped after the fact. This was just in a regular washing machine on regular cycle on cold. Ours wasn't very expensive - I think it was from a school or something. From then on though it always looked like the logo kind of just sat on the fabric. Honestly, my kids didn't care. And ours hadn't said to handwash so I hadn't even bothered.
That could be why they don't want you to 'ring' it out either. It could be the twisting of the logo - might affect it somehow.
Logos definitely don't always do well in the dryer. I've had them kind of melt before. Not so that it's that noticeable, but the texture gets altered. Or they look kind of cracked.
Personally - I'd hand wash it (you can buy a bigger tub if you need to from the dollar store) and do as Marda says - you lay if flat on a towel and form to the right shape, and then you gently roll up like a sponge roll type cake. Hopefully the logo wouldn't be affected too much. Go slow and maybe check it. Then I'd remove from towel and just stick on one of those mesh sweater racks that allow clothes to dry faster.
I find these days a lot of clothes will say "hand wash only". They are just cotton but will come with those instructions. I almost wonder if it's just because fabrics are cheaper these days and stores don't want to be responsible for returns if you didn't follow the directions or something.
The other thing I do - is I will use our delicate wash cycle on cold - with a special detergent meant for cold water - and I throw the item in a laundry/gentle bag - one of those mesh zip up bags. I don't find it always gets super clean (as kind of folded up in bag) but it ensures that clothes aren't getting stretched out of shape.
Good luck :) I washed my son's prized ball cap recently and it lost it's shape. I found a YouTube video that helped me make it better - what a pain over a piece of clothing. But if you ruin their clothing ...