Firstly, if the bag is actually in the duty free shop (World Duty Free), rather than a separate branded concession shop, WDF have stores in each terminal so the product range will be very similar in each place. But, if I recall, there are very few, if any bags in the main DF Stores as these major on liquors, tobaccos, perfumes and cosmetics. I think you are getting confused with a general all brand store.

That said, you could in theory get to the store airside in T5, before you transfer to T1, but you are going to waste a lot of time.

Firstly, sort out the actual store your bag is sold in, then check if this store is also in T1, then contact them. T1 has a pretty good selection of stores, not much difference to T5 all the same brands are there.

As stated, you should be able to check through your bags as transit baggage.