My son is 11 months old and has LI. He has a problem with overheating too. We lived in Phoenix, AZ until mid-June and I don't know how he would have survived the summer had we stayed there. Walking from the car into the store (takes all of one minute!) he turned bright red and it took me quite a while to cool him down. We now live in southern CA and it is soooo much better for him here in that respect.
I take the kids on walks or to the park or whatever just about every day, so Andrej is in the sun a lot. I try to keep him in the shade, and he's still young so it's easier to do that. However, when I am out and about I have a mister that I use on him to help keep him cool. It came from Costco in a two-pack, and it's basically a water bottle with a hose. If you don't want to spend the money on that, a spray bottle works just as well, though you have to refill more often. The kids love getting sprayed - it's a game to them - and I'll be honest, I've come to appreciate a squirt now and then myself!
Soaking his hat in water seems to help, too. I fully intend on making his life as normal as possible as he grows. Our family is huge into soccer, and I anitcipate that he may want to play too. I plan to do whatever it takes to let him do that, even if I have live in a cooler climate, and to dump buckets of water on him between halves!
Good luck and have fun!
Daughter, Emma born 9/2001 -unaffected
Son, Andrej born 8/2003 - has Lamellar Ichthyosis
Daughter, Tori born 3/2006 - has Lamellar Ichthyosis