You can wire it in wet string as long as it works over the distance voltdrop is the thing to remember, it dont cost much to wire in the correct bell wire so why bodge it ?
I have never used bell wire in 30 years. I normally use alarm cable, Doubling up on all the cores needed, but now i realise i have been bodging it all this time i may have to reconsider.
Assuming the bell push is on the outside of the outside then since we have extended the equipotential zone should we bang a rod in and bond the door handle and letter box?