There's nothing wrong with using a 3rd party battery, for years I have used an Anker 4400mAh battery for my Samsung Galaxy S3. I can't fault it whatsoever, it's been absolutely fantastic and to this day still holds charge well. I would recommend Anker, they're not the most expensive either but have always been reliable for me.
If you've pulled the connector off you're not going to be able to do it yourself unless you can solder (don't quote me on this) I don't fix iPads but when when I've dacidentally pulled off weak connectors I can always resolder them. i.e. iPhone 4/4s battery connector is held on by 4 very weak solder points.