seedlings are much more tolerant to transplants than vegging plants. i transplant all the time and here is what you should do to reduce shock.
get your high quality soil ready - i see why you are transplanting! that mix looks like its mostly bark chips - and fill up your pot to the top. water around and dig a little pit with your hand in the middle. water your seedling so the two different mediums are as close in water pressure to each other as the can, then carefully, with a big spoon or something, scoop out the root ball (you can do this with your hands, but setting it in a big spoon is much easier). make sure the hole in your new soil is deep enough and plop her in. press firmly around the top and try to bury the stem in new soil up to the cotyledons so its in there nice and good. now water the plant in with a mild transplant solution if you have one, if not, just water the crap out of the plant - allowing for at least 10% runoff.
try and preform all transplants in the morning so the plant has the whole day to heal. this will help avoid morning after wilting in weaker plants.
make sure you dont smoke too much before hand too - ive dropped root balls/transplanted plants into their old pot by accident and it sucks to see your plant get shocked. im pretty baked, so i hope that covers everything important. good luck