Go to the window "Server Behaviors" you will see the eCart object there.
To remove UPS, in the same window, go to the shipping tab and click the minus button to remove UPS.
To update the caculation go to the confirm page and highlight the UPS Server Behavior. Then click the minus button to remove it.
There you can add your Recordset to get the correct charge by weight from your database of weights and associated charges.
Then use the Server Behavior "Set Session Value" to store the resulting charge in a sesison variable.
At that point you go back into the cart and add the shipping charge equal to the value of that Session variable. If you aren't familiar with Dreamweaver Server Behaviors or PHP it will be difficult to make updates since there are so many possibilities we cannot document them all.