Middle of a battery replacement but battery connecter outlet is broken

Double check that you did damage the SIP module. Here's a blow up of the connector on the SIP board

If you have lifted the connector off of the SIP module its not likely you'll be able to resolder it down as the solder pads are so small you likely pulled them fully off so there is nothing to solder to! Post a clear picture that we can see what you have now iFixitでの質問に画像を追加する方法 but I wouldn't hold out hope, sorry ;-{

At this point I think its time to throw in the towel. while replacing the SIP unit is possible finding one that is not locked is not easy! Just like an iPhone Apple did a very good job to protect your data and in doing so made repairing some areas quite hard and here replacing close to imposable!

So what to do? Carefully button up the watch with the orginal battery. Making it look like you've not entered, visit an Apple Store to see what it will cost to repair, only say the battery stopped working. I'm assuming the watch comes alive when placed on the charging puck. If not see what they will give you for credit to a new watch.