You can deliver it yourself, but this is not without risk.
As you listed it for delivery eBay will expect you to have uploaded postal tracking proof of delivery. Without this, eBay won't treat it as delivered. Neither will you have the QR code that eBay would have provided as proof of collection, if it was listed with this option. Without either, as only a very occasional seller eBay will probably hold your payment for 21 days as a security precaution.
As the buyer is not paying for delivery you will also need to amend their invoice before they pay, which they must do before delivery. Payment in cash on delivery is no longer allowed.
You will have to trust your buyer, as if they dishonestly claim not to have received the item you would have to refund them in full.
Incidentally, why offer buyers a choice of expensive, or presumably cheaper delivery? Most buyer will automatically ask for the cheapest, knowing that eBay will still require you to refund them in full if the item doesn't arrive, or is damaged. Postal insurance is for your benefit, not the buyer's.
