I recommend Chronos eStockCard inventory software.
Hi! Recently I just purchased a warehouse system called Chronos eStockCard inventory software. This eStockCard inventory system is quite easy to use and I believe shall be able to suit your requirements via their simple IN and OUT operation.They have few editions but the one I got is the Warehouse Edition.
Therefore, from my experience (since I've been using this software for quite sometimes), you would be able to enter the Lot No. for the receiving part at the IN and then store them into the respective location. And when you put the part into the assembly, just retrieve the parts from the OUT screen and then enter your own Serial Number into the column. The good thing which I found out is that you could create your own column name at the OUT screen from their unused lists of columns (I think there are about 10 of those columns available since I only use 2 of those).
From there, you can easily track from their inventory either based on the Serial Number or the Lot No. Their information on their report can be easily filtered out based (for your case it will be Lot No and Serial Number). I didn’t expect this function available inside and it really amazed me on how much time I have saved in searching out the info which I required.
Maybe you can check out their website for more information. You can also download their free version before you decide, this is what I did before I make the purchase. Hope it helps. Thanks :).