Shipping cost incorrect on checkout but correct in payment

hi guys, I need a little help here.

Within checkout form customer must choose where he lives and after to choose if he need delivery or he want pickup for free. (Integrated map with regions and prices for the delivery)

when he got to the last point still the checkout where order summary and price it didn’t show delivery and price for that. But when customer press on pay it will count and show expected price with delivery. So basically functionality works fine as it collect amount normally but confusing customer as on summery 1 cost and payment is another.

also if customer decide to go back to checkout and change to local pickup and gets to the last point already there will be delivery fee 🙈

it very seems like I need to to some Ajax update or some triggering so it will immediately show the cost.

any suggestions ?thanks in advance !

The page I need help with: [log in to see the link]