Is there a tool that searches for a specific BTC amount in a specific date range?
Blockchair allow you to query the bitcoin blockchain with lots of parameter such as date, BTC output amount, fee amount etc...

Here is your request:
https://blockchair.com/bitcoin/transactions?q=time(2013-07-27..2013-08-03),output_total(44500000000)#

Unfortunately it bring no result but you can change the parameters