If you get to the bite quickly enough, AfterBite can work miracles. After the fact, our family uses Solarcaine which includes a mild anesthetic and also cools the skin. In terms of preventing bleeding and scabbing, preventing scratching is key, so anything that reduces the itch at the beginning will help. Also, once a bite is scratched, the healing damage from the scratch will itch again tomorrow. You may also consider an antihistamine cream or a system antihistamine (eg Benedryl) at bedtime, in a child appropriate dose, both to ensure a good night's sleep tonight and to reduce tomorrow's itching and misery caused by scratching.
After some camping trips in which I got covered in bites, I discovered that even the mildest scratching slows healing. Bites I couldn't reach (between my shoulder blades) drove me insane but healed in a day or two. Bites I couldn't reach in public (say, on my stomach) healed in a week or two. Bites I could absent-mindedly scratch (on my wrist) lasted for over a month and were uncomfortable the whole time. I now put a bandaid over any bite simply to prevent myself from scratching it.
Some parents swear that patting a bug bite - reasonably firmly but not a slap - brings all the same relief of scratching but without causing damage. A toddler can be taught to pat instead of to scratch, they say, and I have seen kids who do this. We didn't try it ourselves, though.
One major advantage of a small child is suggestibility. The magical parental medical treatments in our house included "a warm cloth" to be held against a bruise or other sore spot and "a cold cloth" for fever and the like. It is amazing what these items, offered by someone who loves you and says they will help, can in fact help with. For one of my children, who swelled up visibly not only from bug bites but also from Solarcaine, I once said "tell your hand not to swell up like that." With some encouragement, the toddler faced the offending hand and delivered a stern verbal order. Within half an hour the swelling was down, and every time a bug bite started to swell, a quick order (from the owner of the hand, not me) took care of it. I can't say the same is guaranteed to work for you, but it can't hurt to try, right?